A036040 -id:A036040 - cp's OEIS frontend

A099174 Triangle read by rows: coefficients of modified Hermite polynomials.

1, 0, 1, 1, 0, 1, 0, 3, 0, 1, 3, 0, 6, 0, 1, 0, 15, 0, 10, 0, 1, 15, 0, 45, 0, 15, 0, 1, 0, 105, 0, 105, 0, 21, 0, 1, 105, 0, 420, 0, 210, 0, 28, 0, 1, 0, 945, 0, 1260, 0, 378, 0, 36, 0, 1, 945, 0, 4725, 0, 3150, 0, 630, 0, 45, 0, 1, 0, 10395, 0, 17325, 0, 6930, 0, 990, 0, 55, 0, 1

Offset: 0

Ralf Stephan, on a suggestion of Karol A. Penson, Oct 13 2004

Absolute values of A066325.
T(n,k) is the number of involutions of {1,2,...,n}, having k fixed points (0 <= k <= n). Example: T(4,2)=6 because we have 1243,1432,1324,4231,3214 and 2134. - Emeric Deutsch, Oct 14 2006
Riordan array [exp(x^2/2),x]. - Paul Barry, Nov 06 2008
Same as triangle of Bessel numbers of second kind, B(n,k) (see Cheon et al., 2013). - N. J. A. Sloane, Sep 03 2013
The modified Hermite polynomial h(n,x) (as in the Formula section) is the numerator of the rational function given by f(n,x) = x + (n-2)/f(n-1,x), where f(x,0) = 1. - Clark Kimberling, Oct 20 2014
Second lower diagonal T(n,n-2) equals positive triangular numbers A000217 \ {0}. - M. F. Hasler, Oct 23 2014
From James East, Aug 17 2015: (Start)
T(n,k) is the number of R-classes (equivalently, L-classes) in the D-class consisting of all rank k elements of the Brauer monoid of degree n.
For n < k with n == k (mod 2), T(n,k) is the rank (minimal size of a generating set) and idempotent rank (minimal size of an idempotent generating set) of the ideal consisting of all rank <= k elements of the Brauer monoid. (End)
This array provides the coefficients of a Laplace-dual sequence H(n,x) of the Dirac delta function, delta(x), and its derivatives, formed by taking the inverse Laplace transform of these modified Hermite polynomials. H(n,x) = h(n,D) delta(x) with h(n,x) as in the examples and the lowering and raising operators L = -x and R = -x + D = -x + d/dx such that L H(n,x) = n * H(n-1,x) and R H(n,x) = H(n+1,x). The e.g.f. is exp[t H(.,x)] = e^(t^2/2) e^(t D) delta(x) = e^(t^2/2) delta(x+t). - Tom Copeland, Oct 02 2016
Antidiagonals of this entry are rows of A001497. - Tom Copeland, Oct 04 2016
This triangle is the reverse of that in Table 2 on p. 7 of the Artioli et al. paper and Table 6.2 on p. 234 of Licciardi's thesis, with associations to the telephone numbers. - Tom Copeland, Jun 18 2018 and Jul 08 2018
See A344678 for connections to a Heisenberg-Weyl algebra of differential operators, matching and independent edge sets of the regular n-simplices with partially labeled vertices, and telephone switchboard scenarios. - Tom Copeland, Jun 02 2021

			h(0,x) = 1
h(1,x) = x
h(2,x) = x^2 + 1
h(3,x) = x^3 + 3*x
h(4,x) = x^4 + 6*x^2 + 3
h(5,x) = x^5 + 10*x^3 + 15*x
h(6,x) = x^6 + 15*x^4 + 45*x^2 + 15
From _Paul Barry_, Nov 06 2008: (Start)
Triangle begins
   1,
   0,  1,
   1,  0,  1,
   0,  3,  0,  1,
   3,  0,  6,  0,  1,
   0, 15,  0, 10,  0,  1,
  15,  0, 45,  0, 15,  0,  1
Production array starts
  0, 1,
  1, 0, 1,
  0, 2, 0, 1,
  0, 0, 3, 0, 1,
  0, 0, 0, 4, 0, 1,
  0, 0, 0, 0, 5, 0, 1 (End)

Alois P. Heinz, Rows n = 0..150, flattened
M. Artioli, G. Dattoli, S. Licciardi, and S. Pagnutti, Motzkin Numbers: an Operational Point of View, arXiv:1703.07262 [math.CO], 2017.
Paul Barry, Riordan array, orthogonal polynomials as moments, and Hankel transforms, arXiv:1102.0921 [math.CO], 2011.
G.-S. Cheon, J.-H. Jung and L. W. Shapiro, Generalized Bessel numbers and some combinatorial settings, Discrete Math., 313 (2013), 2127-2138.
T. Copeland, Juggling Zeros in the Matrix (Example II), 2020.
James East and Robert D. Gray, Diagram monoids and Graham-Houghton graphs: idempotents and generating sets of ideals, arXiv:1404.2359 [math.GR], 2014. See Theorem 8.4 and Table 7. - _James East_, Aug 17 2015
A. Horzela, P. Blasiak, G. E. H. Duchamp, K. A. Penson and A. I. Solomon, A product formula and combinatorial field theory, arXiv:quant-ph/0409152, 2004.
Alexander Kreinin, Combinatorial Properties of Mills' Ratio, arXiv:1405.5852 [math.CO], 2014. See Table 2. - _N. J. A. Sloane_, May 29 2014
S. Licciardi, Umbral Calculus, a Different Mathematical Language, arXiv:1803.03108 [math.CA], 2018.
R. Paris, A uniform asymptotic expansion for the incomplete gamma function, Journal of Computational and Applied Mathematics, 148 (2002), p. 223-239 (See p. 329 and A137286. From _Tom Copeland_, Jan 03 2016).
R. Sazdanovic, A categorification of the polynomial ring, slide presentation, 2011
S. Yang and Z. Qiao, The Bessel numbers and Bessel matrices, Jrn. Math. Rsch. and Exposition, July 2011, Vol. 31, No. 4, pp.627-636. DOI:10.3770/j.issn:1000-341X.2011.04.006.

Row sums (polynomial values at x=1) are A000085.
Polynomial values: A005425 (x=2), A202834 (x=3), A202879(x=4).
Cf. A000217, A001147, A059343, A060821, A066325.
Cf. A000384, A014105, A034839, A049403, A096713, A100861, A104556, A122848, A130757, A176230, A176231.
Cf. A137286.
Cf. A001497.
Cf. A111062, A344678.

T:=proc(n,k) if n-k mod 2 = 0 then n!/2^((n-k)/2)/((n-k)/2)!/k! else 0 fi end: for n from 0 to 12 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form; Emeric Deutsch, Oct 14 2006

nn=10;a=y x+x^2/2!;Range[0,nn]!CoefficientList[Series[Exp[a],{x,0,nn}],{x,y}]//Grid  (* Geoffrey Critzer, May 08 2012 *)
H[0, x_] = 1; H[1, x_] := x; H[n_, x_] := H[n, x] = x*H[n-1, x]-(n-1)* H[n-2, x]; Table[CoefficientList[H[n, x], x], {n, 0, 11}] // Flatten // Abs (* Jean-François Alcover, May 23 2016 *)
T[ n_, k_] := If[ n < 0, 0, Coefficient[HermiteH[n, x I/Sqrt[2]] (Sqrt[1/2]/I)^n, x, k]]; (* Michael Somos, May 10 2019 *)

T(n,k)=if(k<=n && k==Mod(n,2), n!/k!/(k=(n-k)/2)!>>k) \\ M. F. Hasler, Oct 23 2014

import sympy
from sympy import Poly
from sympy.abc import x, y
def H(n, x): return 1 if n==0 else x if n==1 else x*H(n - 1, x) - (n - 1)*H(n - 2, x)
def a(n): return [abs(cf) for cf in Poly(H(n, x), x).all_coeffs()[::-1]]
for n in range(21): print(a(n)) # Indranil Ghosh, May 26 2017

def Trow(n: int) -> list[int]:
    row: list[int] = [0] * (n + 1); row[n] = 1
    for k in range(n - 2, -1, -2):
        row[k] = (row[k + 2] * (k + 2) * (k + 1)) // (n - k)
    return row  # Peter Luschny, Jan 08 2023

def A099174_triangle(dim):
    M = matrix(ZZ,dim,dim)
    for n in (0..dim-1): M[n,n] = 1
    for n in (1..dim-1):
        for k in (0..n-1):
            M[n,k] = M[n-1,k-1]+(k+1)*M[n-1,k+1]
    return M
A099174_triangle(9)  # Peter Luschny, Oct 06 2012

h(k, x) = (-I/sqrt(2))^k * H(k, I*x/sqrt(2)), H(n, x) the Hermite polynomials (A060821, A059343).
T(n,k) = n!/(2^((n-k)/2)*((n-k)/2)!k!) if n-k >= 0 is even; 0 otherwise. - Emeric Deutsch, Oct 14 2006
G.f.: 1/(1-x*y-x^2/(1-x*y-2*x^2/(1-x*y-3*x^2/(1-x*y-4*x^2/(1-... (continued fraction). - Paul Barry, Apr 10 2009
E.g.f.: exp(y*x + x^2/2). - Geoffrey Critzer, May 08 2012
Recurrence: T(0,0)=1, T(0,k)=0 for k>0 and for n >= 1 T(n,k) = T(n-1,k-1) + (k+1)*T(n-1,k+1). - Peter Luschny, Oct 06 2012
T(n+2,n) = A000217(n+1), n >= 0. - M. F. Hasler, Oct 23 2014
The row polynomials P(n,x) = (a. + x)^n, umbrally evaluated with (a.)^n = a_n = aerated A001147, are an Appell sequence with dP(n,x)/dx = n * P(n-1,x). The umbral compositional inverses (cf. A001147) of these polynomials are given by the same polynomials signed, A066325. - Tom Copeland, Nov 15 2014
From Tom Copeland, Dec 13 2015: (Start)
The odd rows are (2x^2)^n x n! L(n,-1/(2x^2),1/2), and the even, (2x^2)^n n! L(n,-1/(2x^2),-1/2) in sequence with n= 0,1,2,... and L(n,x,a) = Sum_{k=0..n} binomial(n+a,k+a) (-x)^k/k!, the associated Laguerre polynomial of order a. The odd rows are related to A130757, and the even to A176230 and A176231. Other versions of this entry are A122848, A049403, A096713 and A104556, and reversed A100861, A144299, A111924. With each non-vanishing diagonal divided by its initial element A001147(n), this array becomes reversed, aerated A034839.
Create four shift and stretch matrices S1,S2,S3, and S4 with all elements zero except S1(2n,n) = 1 for n >= 1, S2(n,2n) = 1 for n >= 0, S3(2n+1,n) = 1 for n >= 1, and S4(n,2n+1) = 1 for n >= 0. Then this entry's lower triangular matrix is T = Id + S1 * (A176230-Id) * S2 + S3 * (unsigned A130757-Id) * S4 with Id the identity matrix. The sandwiched matrices have infinitesimal generators with the nonvanishing subdiagonals A000384(n>0) and A014105(n>0).
As an Appell sequence, the lowering and raising operators are L = D and R = x + dlog(exp(D^2/2))/dD = x + D, where D = d/dx, L h(n,x) = n h(n-1,x), and R h(n,x) = h(n+1,x), so R^n 1 = h(n,x). The fundamental moment sequence has the e.g.f. e^(t^2/2) with coefficients a(n) = aerated A001147, i.e., h(n,x) = (a. + x)^n, as noted above. The raising operator R as a matrix acting on o.g.f.s (formal power series) is the transpose of the production matrix P below, i.e., (1,x,x^2,...)(P^T)^n (1,0,0,...)^T = h(n,x).
For characterization as a Riordan array and associations to combinatorial structures, see the Barry link and the Yang and Qiao reference. For relations to projective modules, see the Sazdanovic link.
(End)
From the Appell formalism, e^(D^2/2) x^n = h_n(x), the n-th row polynomial listed below, and e^(-D^2/2) x^n = u_n(x), the n-th row polynomial of A066325. Then R = e^(D^2/2) * x * e^(-D^2/2) is another representation of the raising operator, implied by the umbral compositional inverse relation h_n(u.(x)) = x^n. - Tom Copeland, Oct 02 2016
h_n(x) = p_n(x-1), where p_n(x) are the polynomials of A111062, related to the telephone numbers A000085. - Tom Copeland, Jun 26 2018
From Tom Copeland, Jun 06 2021: (Start)
In the power basis x^n, the matrix infinitesimal generator M = A132440^2/2, when acting on a row vector for an o.g.f., is the matrix representation for the differential operator D^2/2.
e^{M} gives the coefficients of the Hermite polynomials of this entry.
The only nonvanishing subdiagonal of M, the second subdiagonal (1,3,6,10,...), gives, aside from the initial 0, the triangular numbers A000217, the number of edges of the n-dimensional simplices with (n+1) vertices. The perfect matchings of these simplices are the aerated odd double factorials A001147 noted above, the moments for the Hermite polynomials.
The polynomials are also generated from A036040 with x[1] = x, x[2] = 1, and the other indeterminates equal to zero. (End)

A134685 Irregular triangle read by rows: coefficients C(j,k) of a partition transform for direct Lagrange inversion.

Original entry on oeis.org

1, -1, 3, -1, -15, 10, -1, 105, -105, 10, 15, -1, -945, 1260, -280, -210, 35, 21, -1, 10395, -17325, 6300, 3150, -280, -1260, -378, 35, 56, 28, -1, -135135, 270270, -138600, -51975, 15400, 34650, 6930, -2100, -1575, -2520, -630, 126, 84, 36, -1

Offset: 1

Tom Copeland, Jan 26 2008, Sep 13 2008

Let f(t) = u(t) - u(0) = Ev[exp(u.* t) - u(0)] = log{Ev[(exp(z.* t))/z_0]} = Ev[-log(1- a.* t)], where the operator Ev denotes umbral evaluation of the umbral variables u., z. or a., e.g., Ev[a.^n + a.^m] = a_n + a_m . The relation between z_n and u_n is given in reference in A127671 and u_n = (n-1)! * a_n .
If u_1 is not equal to 0, then the compositional inverse for these expressions is given by g(t) = Sum_{j>=1} P(j,t) where, with u_n denoted by (n') for brevity,
P(1,t) = (1')^(-1) * [ 1 ] * t
P(2,t) = (1')^(-3) * [ -(2') ] * t^2 / 2!
P(3,t) = (1')^(-5) * [ 3 (2')^2 - (1')(3') ] * t^3 / 3!
P(4,t) = (1')^(-7) * [ -15 (2')^3 + 10 (1')(2')(3') - (1')^2 (4') ] * t^4 / 4!
P(5,t) = (1')^(-9) * [ 105 (2')^4 - 105 (1') (2')^2 (3') + 10 (1')^2 (3')^2 + 15 (1')^2 (2') (4') - (1')^3 (5') ] * t^5 / 5!
P(6,t) = (1')^(-11) * [ -945 (2')^5 + 1260 (1') (2')^3 (3') - 280 (1')^2 (2') (3')^2 - 210 (1')^2 (2')^2 (4') + 35 (1')^3 (3')(4') + 21 (1')^3 (2')(5') - (1')^4 (6') ] * t^6 / 6!
P(7,t) = (1')^(-13) * [ 10395 (2')^6 - 17325 (1') (2')^4 (3') +  (1')^2 [ 6300 (2')^2 (3')^2 + 3150  (2')^3 (4')] - (1')^3 [280 (3')^3 + 1260  (2')(3')(4') + 378  (2')^2(5')] + (1')^4 [35 (4')^2 + 56 (3')(5') + 28 (2')(6')]  - (1')^5 (7') ] * t^7 / 7!
P(8,t) = (1')^(-15) * [ -135135 (2')^7 + 270270 (1') (2')^5 (3') -  (1')^2 [ 138600 (2')^3 (3')^2 +  51975 (2')^4 (4')] + (1')^3 [15400 (2')(3')^3 + 34650  (2')^2(3')(4') + 6930  (2')^3(5')] - (1')^4 [2100 (3')^2(4') + 1575 (2')(4')^2 + 2520 (2')(3')(5') + 630 (2')^2(6') ]  + (1')^5 [126 (4')(5') + 84 (3')(6') + 36 (2')(7')] - (1')^6 (8') ] * t^8 / 8!
...
Substituting ((m-1)') for (m') in each partition and ignoring the (0') factors, the partitions in the brackets of P(n,t) become those of n-1 listed in Abramowitz and Stegun on page 831 (in the reversed order) and the number of partitions in P(n,t) is given by A000041(n-1).
Combinatorial interpretations are given in the link.
From Tom Copeland, Jul 10 2018: (Start)
Coefficients occurring in prolongation for the special Euclidean group SE(2) and special affine group SA(2) in the Olver presentation on moving frames (MFP) in slides 33 and 42. These are a result of applying an iterated derivative of the form h(x)d/dx = d/dy as in this entry (more generally as g(x) d/dx as discussed in A145271). See also p. 6 of Olver's paper on contact forms, but note that the 12 should be a 15 in the formula for the compositional inverse of S(t).
Change variables in the MFP to obtain connections to the partition polynomials Prt_n = n! * P(n,1) above. Let delta and beta in the formulas for the equi-affine curves in MFP be L and B, respectively, and D_y = (1/(L-B*u_x)) d/dx = (1/w_x) d/dx. Then v_(yy) = (1/B) [-w_(xx)/(w_x)^3] in MFP (there is an overall sign error in MFP for v_(yy) and higher derivatives w.r.t. y), and (d/dy)^n v = v_n = (1/B)* [(1/w_1)*(d/dx)]^(n-2) [-w_2/(w_1)^3] for n > 1, with w_n = (d/dx)^n w. Consequently, in the partition polynomials Prt_n for n > 1 here substitute (n') = -B*u_n = w_n for n > 1 and (1')  = L-B*u_1 = w_1, where u_n = (d/dx)^n u, and then divide by B. For example, v_4 = (1/B)*Prt_4 = (1/B)*4!*P(4,1) = (1/B) (L-B*u_n)^(-7) [-15*(-B*u_2)^3 + 10 (L-B*u_1)(-B*u_2)(-B*u_3) - (L-B*u_1)^2 (-B*u_4)], agreeing with v_4 in MFP except for the overall sign.
For the SE(2) transformation formulas in MFP, let w_x = cos(phi) + sin(phi)*u_x, and then the same transformations apply as above with cos(phi) and sin(phi) substituted for L and -B, respectively. (End)

			Examples and checks:
1) Let u_1 = -1 and u_n = 1 for n>1,
then f(t) = exp(u.*t) - u(0) = exp(t)-2t-1
and g(t) = [e.g.f. of signed A000311];
therefore, the row sums of unsigned [C(j,k)] are A000311 =
(0,1,1,4,26,236,2752,...) = (0,-P(1,1),2!*P(2,1),-3!*P(3,1),4!*P(4,1),...).
2) Let u_1 = -1 and u_n = (n-1)! for n>1,
then f(t) = -log(1-t)-2t
and g(t) = [e.g.f. of signed (0,A032188)]
with (0,A032188) = (0,1,1,5,41,469,6889,...) = (0,-P(1,1),2!*P(2,1),-3!P(3,1),...).
3) Let u_1 = -1 and u_n = (-1)^n (n-2)! for n>1, then
f(t) = (1+t) log(1+t) - 2t
and g(t) = [e.g.f. of signed (0,A074059)]
with (0,A074059) = (0,1,1,2,7,34,213,...) = (0,-P(1,1),2!*P(2,1),-3!*P(3,1),...).
4) Let u_1 = 1, u_2 = -1 and u_n = 0 for n>2,
then f(t) = t(1-t/2)
and g(t) = [e.g.f. of (0,A001147)] = 1 - (1-2t)^(1/2)
with (0,A001147) = (0,1,1,3,15,105,945...) =(0,P(1,1),2!*P(2,1),3!*P(3,1),...).
5) Let u_1 = 1, u_2 = -2 and u_n = 0 for n>2,
then f(t)= t(1-t)
and g(t) = t * [o.g.f. of A000108] = [1 - (1-4t)^(1/2)] / 2
with (0,A000108) = (0,1,1,2,5,14,42,...) = (0,P(1,1),P(2,1),P(3,1),...).
.
From _Peter Luschny_, Feb 19 2021: (Start)
Triangle starts:
 [1]  1;
 [2] -1;
 [3]  3,     -1;
 [4] -15,     10,    -1;
 [5]  105,   -105,   [10, 15],  -1;
 [6] -945,    1260,  [-280, -210], [35, 21],  -1;
 [7]  10395, -17325, [6300, 3150], [-280, -1260, -378], [35, 56, 28], -1;
 [8] -135135, 270270, [-138600, -51975], [15400, 34650, 6930], [-2100, -1575, -2520, -630], [126, 84, 36], -1
The coefficients can be seen as a refinement of the Ward numbers: Let R(n, k) = Sum T(n, k), where the sum collects adjacent terms with equal sign, as indicated by the square brackets in the table, then R(n+1, k+1) = (-1)^(n-k)*W(n, k), where W(n, k) are the Ward numbers A181996, for n >= 0 and 0 <= k <= n-1.  (End)

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 831.
D. S. Alexander, A History of Complex Dynamics: From Schröder to Fatou to Julia, Friedrich Vieweg & Sohn, 1994, p. 10.
J. Riordan, Combinatorial Identities, Robert E. Krieger Pub. Co., 1979, (unsigned partition polynomials in Table 5.2 on p. 181, but may have errors).

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
O. Arnaldsson, Élie Cartan’s Theory of Moving Frames, slide presentation, 2014.
Tom Copeland, Generators, Inversion, and Matrix, Binomial, and Integral Transforms, 2015.
Tom Copeland, Important formulas in combinatorics, MathOverflow answer, 2015.
Tom Copeland, Compositional inversion and generating functions in algebraic geometry, MathOverflow question, 2014.
Tom Copeland, Compositional inverse pairs, the inviscid Burgers-Hopf equation, and the Stasheff associahedra, 2014.
Tom Copeland, Lagrange a la Lah, 2011.
Tom Copeland, Short Note on Lagrange Inversion, 2008.
Tom Copeland, Formal group laws and binomial Sheffer sequences, 2018.
Bishal Deb and Alan D. Sokal, Higher-order Stirling cycle and subset triangles: Total positivity, continued fractions and real-rootedness, arXiv:2507.18959 [math.CO], 2025. See p. 32.
Hector Figueroa and Jose M. Gracia-Bondia, Combinatorial Hopf algebras in quantum field theory I, arXiv:0408145 [hep-th], 2005, (p. 38).
Ezra Getzler, The semi-classical approximation for modular operads, arXiv:alg-geom/9612005, 1996 (see p. 2).
Peter J. Olver, The canonical contact form.
Peter J. Olver, Moving Frames, slide presentation, 2009.
Andrew Elvey Price and Alan D. Sokal, Phylogenetic trees, augmented perfect matchings, and a Thron-type continued fraction (T-fraction) for the Ward polynomials, arXiv:2001.01468 [math.CO], 2020.
Jair Patrick Taylor, Formal group laws and hypergraph colorings, doctoral thesis, Univ. of Wash., 2016, p. 95.

Cf. A145271, (A134991, A019538) = (reduced array, associated g(x)).
Cf. A000108, A000311, A001147, A032188, A049019, A074059, A129185, A133314, A139605, A036040.
Cf. A181996 (Ward numbers).

rows[n_] := {{1}}~Join~Module[{h = 1/(1 + Sum[u[k] y^k/k!, {k, n-1}] + O[y]^n), g = y, r}, r = Reap[Do[g = h D[g, y]; Sow[Expand[Normal@g /. {y -> 0}]], {k, n}]][[2, 1, ;;]]; Table[Coefficient[r[[k]], Product[u[t], {t, p}]], {k, 2, n}, {p, Reverse@Sort[Sort /@ IntegerPartitions[k-1]]}]];
rows[8] // Flatten (* Andrei Zabolotskii, Feb 19 2024 *)

The bracketed partitions of P(n,t) are of the form (u_1)^e(1) (u_2)^e(2) ... (u_n)^e(n) with coefficients given by (-1)^(n-1+e(1)) * [2*(n-1)-e(1)]! / [2!^e(2)*e(2)!*3!^e(3)*e(3)! ... n!^e(n)*e(n)! ].
From Tom Copeland, Sep 05 2011: (Start)
Let h(t) = 1/(df(t)/dt)
  = 1/Ev[u.*exp(u.*t)]
  = 1/(u_1+(u_2)*t+(u_3)*t^2/2!+(u_4)*t^3/3!+...),
  an e.g.f. for the partition polynomials of A133314
  (signed A049019) with an index shift.
  Then for the partition polynomials of A134685,
  n!*P(n,t) = ((t*h(y)*d/dy)^n) y evaluated at y=0,
  and the compositional inverse of f(t) is
  g(t) = exp(t*h(y)*d/dy) y evaluated at y=0.
  Also, dg(t)/dt = h(g(t)). (Cf. A000311 and A134991)(End)
From Tom Copeland, Oct 30 2011: (Start)
With exp[x* PS(.,t)] = exp[t*g(x)]=exp[x*h(y)d/dy] exp(t*y) eval. at y=0, the raising/creation and lowering/annihilation operators
defined by R PS(n,t)=PS(n+1,t) and L PS(n,t)= n*PS(n-1,t) are
R = t*h(d/dt) = t * 1/[u_1+(u_2)*d/dt+(u_3)*(d/dt)^2/2!+...],  and
L = f(d/dt)=(u_1)*d/dt+(u_2)*(d/dt)^2/2!+(u_3)*(d/dt)^3/3!+....
Then  P(n,t) = (t^n/n!) dPS(n,z)/dz  eval. at z=0. (Cf. A139605, A145271, and link therein to Mathemagical Forests for relation to planted trees on p. 13.) (End)
The bracketed partition polynomials of P(n,t) are also given by (d/dx)^(n-1) 1/[u_1 + u_2 * x/2! + u_3 * x^2/3! + ... + u_n * x^(n-1)/n!]^n evaluated at x=0. - Tom Copeland, Jul 07 2015
Equivalent matrix computation: Multiply the m-th diagonal (with m=1 the index of the main diagonal) of the lower triangular Pascal matrix by u_m = (d/dx)^m f(x) evaluated at x=0 to obtain the matrix UP with UP(n,k) = binomial(n,k) u_{n+1-k}. Then P(n,t) = (1, 0, 0, 0, ...) [UP^(-1) * S]^(n-1) FC * t^n/n!, where S is the shift matrix A129185, representing differentiation in the basis x^n//n!, and FC is the first column of UP^(-1), the inverse matrix of UP. These results follow from A145271 and A133314. - Tom Copeland, Jul 15 2016
Also, P(n,t) = (1, 0, 0, 0, ...) [UP^(-1) * S]^n (0, 1, 0, ..)^T * t^n/n! in agreement with A139605. - Tom Copeland, Aug 27 2016
From Tom Copeland, Sep 20 2016: (Start)
Let PS(n,u1,u2,...,un) = P(n,t) / (t^n/n!), i.e., the square-bracketed part of the partition polynomials in the expansion for the inverse in the comment section, with u_k = uk.
Also let PS(n,u1=1,u2,...,un) = PB(n,b1,b2,...,bK,...) where each bK represents the partitions of PS, with u1 = 1, that have K components or blocks, e.g., PS(5,1,u2,...,u5) = PB(5,b1,b2,b3,b4) = b1 + b2 + b3 + b4 with b1 = -u5, b2 = 15 u2 u4 + 10 u3^2, b3 = -105 u2^2 u3,  and b4 = 105 u2^4.
The relation between solutions of the inviscid Burgers' equation and compositional inverse pairs (cf. link and A086810) implies, for n > 2,  PB(n, 0 * b1, 1 * b2,..., (K-1) * bK, ...) = (1/2) * Sum_{k = 2..n-1} binomial(n+1,k) * PS(n-k+1,u_1=1,u_2,...,u_(n-k+1)) * PS(k,u_1=1,u_2,...,u_k).
For example, PB(5,0 * b1, 1 * b2, 2 * b3, 3 * b4) = 3 * 105 u2^4 - 2 * 105 u2^2 u3 + 1 * 15 u2 u4 + 1 * 10 u3^2 - 0 * u5 = 315 u2^4 - 210 u2^2 u3 + 15 u2 u4 + 10 u3^2 = (1/2) [2 * 6!/(4!*2!) * PS(2,1,u2) * PS(4,1,u2,...,u4) + 6!/(3!*3!) * PS(3,1,u2,u3)^2] = (1/2) * [ 2 * 6!/(4!*2!)  * (-u2) (-15 u2^3 + 10 u2 u3 - u4) + 6!/(3!*3!) * (3 u2^2 - u3)^2].
Also, PB(n,0*b1,1*b2,...,(K-1)*bK,...) =  d/dt t^(n-2)*PS(n,u1=1/t,u2,...,un)|{t=1} = d/dt (1/t)*PS(n,u1=1,t*u2,...,t*un)|{t=1}.
(End)
A recursion relation for computing each partition polynomial of this entry from the lower order polynomials and the coefficients of the Bell polynomials of A036040 is presented in the blog entry "Formal group laws and binomial Sheffer sequences." - Tom Copeland, Feb 06 2018

P(7,t) and P(8,t) data added by Tom Copeland, Jan 14 2016

Terms in rows 5-8 reordered by Andrei Zabolotskii, Feb 19 2024

A130561 Numbers associated to partitions, used for combinatoric interpretation of Lah triangle numbers A105278; elementary Schur polynomials / functions.

Original entry on oeis.org

1, 2, 1, 6, 6, 1, 24, 24, 12, 12, 1, 120, 120, 120, 60, 60, 20, 1, 720, 720, 720, 360, 360, 720, 120, 120, 180, 30, 1, 5040, 5040, 5040, 5040, 2520, 5040, 2520, 2520, 840, 2520, 840, 210, 420, 42, 1, 40320, 40320, 40320, 40320, 20160, 20160, 40320, 40320, 20160

Offset: 1

Wolfdieter Lang, Jul 13 2007

The order of this array is according to the Abramowitz-Stegun (A-St) ordering of partitions (see A036036).
The row lengths sequence is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ...].
These numbers are similar to M_0, M_1, M_2, M_3, M_4 given in A111786, A036038, A036039, A036040, A117506, respectively.
Combinatorial interpretation: a(n,k) counts the sets of lists (ordered subsets) obtained from partitioning the set {1..n}, with the lengths of the lists given by the k-th partition of n in A-St order. E.g., a(5,5) is computed from the number of sets of lists of lengths [1^1,2^2] (5th partition of 5 in A-St order). Hence a(5,5) = binomial(5,2)*binomial(3,2) = 5!/(1!*2!) = 60 from partitioning the numbers 1,2,...,5 into sets of lists of the type {[.],[..],[..]}.
This array, called M_3(2), is the k=2 member of a family of partition arrays generalizing A036040 which appears as M_3 = M_3(k=1). S2(2) = A105278 (unsigned Lah number triangle) is related to M_3(2) in the same way as S2(1), the Stirling2 number triangle, is related to M_3(1). - Wolfdieter Lang, Oct 19 2007
Another combinatorial interpretation: a(n,k) enumerates unordered forests of increasing binary trees which are described by the k-th partition of n in the Abramowitz-Stegun order. - Wolfdieter Lang, Oct 19 2007
A relation between partition polynomials formed from these "refined Lah numbers" and Lagrange inversion for an o.g.f. is presented in the link "Lagrange a la Lah" along with an e.g.f. and an umbral binary operator tree representation. - Tom Copeland, Apr 12 2011
With the indeterminates (x_1,x_2,x_3,...) = (t,-c_2*t,-c_3*t,...) with c_n >0, umbrally P(n,a.) = P(n,t)|{t^n = a_n} = 0 and P(j,a.)P(k,a.) = P(j,t)P(k,t)|{t^n =a_n} = d_{j,k} >= 0 is the coefficient of x^j/j!*y^k/k! in the Taylor series expansion of the formal group law FGL(x,y) = f[f^{-1}(x)+f^{-1}(y)], where a_n are the inversion partition polynomials for calculating f(x) from the coefficients of the series expansion of f^{-1}(x) given in A133437. - Tom Copeland, Feb 09 2018
Divided by n!, the row partition polynomials are the elementary homogeneous Schur polynomials presented on p. 44 of the Bracci et al. paper. - Tom Copeland, Jun 04 2018
Also presented (renormalized) as the Schur polynomials on p. 19 of the Konopelchenko and Schief paper with associations to differential operators related to the KP hierarchy. - Tom Copeland, Nov 19 2018
Through equation 4.8 on p. 26 of the Arbarello reference, these polynomials appear in the Hirota bilinear equations 4.7 related to tau-function solutions of the KP hierarchy. - Tom Copeland, Jan 21 2019
These partition polynomials appear as Feynman amplitudes in their Bell polynomial guise (put x_n = n!c_n in A036040 for the indeterminates of the Bell polynomials) in Kreimer and Yeats and Balduf (e.g., p. 27). - Tom Copeland, Dec 17 2019
From Tom Copeland, Oct 15 2020: (Start)
With a_n = n! * b_n = (n-1)! * c_n for n  > 0, represent a function with f(0) = a_0 = b_0 = 1 as an
A) exponential generating function (e.g.f), or formal Taylor series: f(x) = e^{a.x} = 1 + Sum_{n > 0} a_n * x^n/n!
B) ordinary generating function (o.g.f.), or formal power series: f(x)  = 1/(1-b.x) = 1 + Sum_{n > 0}  b_n * x^n
C) logarithmic generating function (l.g.f): f(x) = 1 - log(1 - c.x) = 1 + Sum_{n > 0}  c_n * x^n /n.
Expansions of log(f(x)) are given in
I) A127671 and A263634 for the e.g.f: log[ e^{a.*x} ] =  e^{L.(a_1,a_2,...)x} = Sum_{n > 0} L_n(a_1,...,a_n) * x^n/n!, the logarithmic polynomials, cumulant expansion polynomials
II) A263916 for the o.g.f.: log[ 1/(1-b.x) ] =  log[ 1 - F.(b_1,b_2,...)x ] = -Sum_{n > 0} F_n(b_1,...,b_n) * x^n/n, the Faber polynomials.
Expansions of exp(f(x)-1) are given in
III) A036040 for an e.g.f: exp[ e^{a.x} - 1 ] = e^{BELL.(a_1,...)x}, the Bell/Touchard/exponential partition polynomials, a.k.a. the Stirling partition polynomials of the second kind
IV) A130561 for an o.g.f.: exp[ b.x/(1-b.x) ] = e^{LAH.(b.,...)x}, the Lah partition polynomials
V) A036039 for an l.g.f.: exp[ -log(1-c.x) ] =  e^{CIP.(c_1,...)x}, the cycle index polynomials of the symmetric groups S_n, a.k.a. the Stirling partition polynomials of the first kind.
Since exp and log are a compositional inverse pair, one can extract the indeterminates of the log set of partition polynomials from the exp set and vice versa. For a discussion of the relations among these polynomials and the combinatorics of connected and disconnected graphs/maps, see Novak and LaCroix on classical moments and cumulants and the two books on statistical mechanics referenced in A036040. (End)
These partition polynomials are referred to as Schur functions by Segal and Wilson, who present associations with Plucker coordinates, Grassmannians, and the tau functions of the KdV hierarchy. See pages 51 and 61. - Tom Copeland, Jan 08 2022

			Triangle starts:
  [  1];
  [  2,   1];
  [  6,   6,   1];
  [ 24,  24,  12, 12,  1];
  [120, 120, 120, 60, 60, 20, 1];
  ...
a(5,6) = 20 = 5!/(3!*1!) because the 6th partition of 5 in A-St order is [1^3,2^1].
a(5,5) = 60 enumerates the unordered [1^1,2^2]-forest with 5 vertices (including the three roots) composed of three such increasing binary trees: 5*((binomial(4,2)*2)*(1*2))/2! = 5*12 = 60.

E. Arbarello, "Sketches of KdV", Contemp. Math. 312 (2002), p. 9-69.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
P. Balduf, The propagator and diffeomorphisms of an interacting field theory, Master's thesis, submitted to the Institut für Physik, Mathematisch-Naturwissenschaftliche Fakultät, Humboldt-Universtität, Berlin, 2018.
F. Bracci, M. Contreras, S. Díaz-Madrigal, and A. Vasil'ev, Classical and stochastic Löwner-Kufarev equations, arXiv:1309.6423 [math.CV], 2013.
T. Copeland, Generators, Inversion, and Matrix, Binomial, and Integral Transforms
T. Copeland, Lagrange a la Lah
T. Copeland, Formal group laws and binomial Sheffer sequences, 2018.
T. Copeland, In the Realm of Shadows: Umbral inverses and associahedra, noncrossing partitions, symmetric polynomials, and similarity transforms, 2019.
G. Duchamp, Important formulas in combinatorics: The exponential formula, a Mathoverflow answer, 2015
T. Ernst, A history of the q-calculus and a new method, 2000.
B. Konopelchenko, Quantum deformations of associative algebras and integrable systems, arXiv:0802.3022 [nlin.SI], 2008.
D. Kreimer and K. Yeats, Diffeomorphisms of quantum fields, arXiv:1610.01837 [math-ph], 2016.
Wolfdieter Lang, Combinatorial Interpretation of Generalized Stirling Numbers, J. Int. Seqs. Vol. 12 (2009) 09.3.3.
Wolfdieter Lang, First 10 rows and more.
J. Novak and M. LaCroix, Three lectures on free probability, arXiv:1205.2097 [math.CO], 2012.
G. Segal and G. Wilson, Loop groups and equations of KdV type, Publications mathématiques de l’I.H.É.S., tome 61, 1985, p. 5-65.

Cf. A105278 (unsigned Lah triangle |L(n, m)|) obtained by summing the numbers for given part number m.
Cf. A000262 (row sums), identical with row sums of unsigned Lah triangle A105278.
A134133(n, k) = A130561(n, k)/A036040(n, k) (division by the M_3 numbers). - Wolfdieter Lang, Oct 12 2007
Cf. A036039, A263634, A263916.
Cf. A096162.
Cf. A133437.
Cf. A127671.

a(n,k) = n!/(Product_{j=1..n} e(n,k,j)!) with the exponent e(n,k,j) of j in the k-th partition of n in the A-St ordering of the partitions of n. Exponents 0 can be omitted due to 0!=1.
From Tom Copeland, Sep 18 2011: (Start)
Raising and lowering operators are given for the partition polynomials formed from A130561 in the Copeland link in "Lagrange a la Lah Part I" on pp. 22-23.
An e.g.f. for the partition polynomials is on page 3:
  exp[t*:c.*x/(1-c.*x):] = exp[t*(c_1*x + c_2*x^2 + c_3*x^3 + ...)] where :(...): denotes umbral evaluation of the enclosed expression and c. is an umbral coefficient. (End)
From Tom Copeland, Sep 07 2016: (Start)
The row partition polynomials of this array P(n,x_1,x_2,...,x_n), given in the Lang link, are n! * S(n,x_1,x_2,...,x_n), where S(n,x_1,...,x_n) are the elementary Schur polynomials, for which d/d(x_m) S(n,x_1,...,x_n) = S(n-m,x_1,...,x_(n-m)) with S(k,...) = 0 for k < 0, so d/d(x_m) P(n,x_1,...,x_n) = (n!/(n-m)!) P(n-m,x_1,...,x_(n-m)), confirming that the row polynomials form an Appell sequence in the indeterminate x_1 with P(0,...) = 1. See p. 127 of the Ernst paper for more on these Schur polynomials.
With the e.g.f. exp[t * P(.,x_1,x_2,..)] = exp(t*x_1) * exp(x_2 t^2 + x_3 t^3 + ...), the e.g.f. for the partition polynomials that form the umbral compositional inverse sequence U(n,x_1,...,x_n) in the indeterminate x_1 is exp[t * U(.,x_1,x_2,...)] = exp(t*x_1) exp[-(x_2 t^2 + x_3 t^3 + ...)]; therefore, U(n,x_1,x_2,...,x_n) = P(n,x_1,-x_2,.,-x_n), so umbrally P[n,P(.,x_1,-x_2,-x_3,...),x_2,x_3,...,x_n] = (x_1)^n = P[n,P(.,x_1,x_2,...),-x_2,-x_3,...,-x_n]. For example, P(1,x_1) = x_1, P2(x_1,x_2) = 2 x_2 + x_1^2, and P(3,x_1,x_2,x_3) = 6 x_3 + 6 x_2 x_1 + x_1^3, then P[3,P(.,x_1,-x_2,...),x_2,x_3] = 6 x_3 + 6 x_2 P(1,x_1) + P(3,x_1,-x_2,-x_3) = 6 x_3 + 6 x_2 x_1 + 6 (-x_3) + 6 (-x_2) x_1 + x_1^3 = x_1^3.
From the Appell formalism, umbrally [P(.,0,x_2,x_3,...) + y]^n = P(n,y,x_2,x_3,...,x_n).
The indeterminates of the partition polynomials can also be extracted using the Faber polynomials of A263916 with -n * x_n = F(n,S(1,x_1),...,S(n,x_1,...,x_n)) = F(n,P(1,x_1),...,P(n,x_1,...,x_n)/n!). Compare with A263634.
Also P(n,x_1,...,x_n) = ST1(n,x_1,2*x_2,...,n*x_n), where ST1(n,...) are the row partition polynomials of A036039.
(End)

Name augmented by Tom Copeland, Dec 08 2022

A162663 Table by antidiagonals, T(n,k) is the number of partitions of {1..(nk)} that are invariant under a permutation consisting of n k-cycles.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 2, 7, 5, 1, 3, 8, 31, 15, 1, 2, 16, 42, 164, 52, 1, 4, 10, 111, 268, 999, 203, 1, 2, 28, 70, 931, 1994, 6841, 877, 1, 4, 12, 258, 602, 9066, 16852, 51790, 4140, 1, 3, 31, 106, 2892, 6078, 99925, 158778, 428131, 21147, 1, 4, 22, 329, 1144, 37778, 70402, 1224579, 1644732, 3827967, 115975

Offset: 0

Franklin T. Adams-Watters, Jul 09 2009

The upper left corner of the array is T(0,1).
Without loss of generality, the permutation can be taken to be (1 2 ... k) (k+1 k+2 ... 2k) ... ((n-1)k+1 (n-1)k+2 ... nk).
Note that it is the partition that is invariant, not the individual parts. Thus for n=k=2 with permutation (1 2)(3 4), the partition 1,3|2,4 is counted; it maps to 2,4|1,3, which is the same partition.

			The table starts:
   1,   1,   1,   1,   1
   1,   2,   2,   3,   2
   2,   7,   8,  16,  10
   5,  31,  42, 111,  70
  15, 164, 268, 931, 602

Alois P. Heinz, Antidiagonals n = 0..140, flattened (first 20 antidiagonals from Franklin T. Adams-Watters)
T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176. [Annotated, scanned copy]
OEIS Wiki, Sorting numbers

Columns: A000110, A002872, A002874, A141003, A036075, A141004, A036077, A141005, A141006, A141007, A036081, A141008, A141009, A141010, A141011.
Rows: A000012, A000005, A162664, A162665.
Cf. A084423, A036040, A036073, A080575.
Main diagonal gives A293850.

with(numtheory):
A:= proc(n, k) option remember; `if`(n=0, 1, add(binomial(n-1, j-1)
       *add(d^(j-1), d=divisors(k))*A(n-j, k), j=1..n))
    end:
seq(seq(A(n, 1+d-n), n=0..d), d=0..12);  # Alois P. Heinz, Oct 29 2015

max = 11; ClearAll[col]; col[k_] := col[k] =  CoefficientList[ Series[ Exp[ Sum[ (Exp[d*x] - 1)/d, {d, Divisors[k]}]], {x, 0, max}], x]*Range[0, max]!; t[n_, k_] := col[k][[n]]; Flatten[ Table[ t[n-k+1, k], {n, 1, max}, {k, n, 1, -1}] ] (* Jean-François Alcover, Aug 08 2012, after e.g.f. *)

amat(n,m)=local(r);r=matrix(n,m,i,j,1);for(k=1,n-1,for(j=1,m,r[k+1,j]=sum (i=1,k,binomial(k-1,i-1)*sumdiv(j,d,r[k-i+1,j]*d^(i-1)))));r
acol(n,k)=local(fn);fn=exp(sumdiv(k,d,(exp(d*x+x*O(x^n))-1)/d));vector(n+ 1,i,polcoeff(fn,i-1)*(i-1)!)

E.g.f. for column k: exp(Sum_{d|k} (exp(d*x) - 1) / d).
Equivalently, column k is the exponential transform of a(n) = Sum_{d|k} d^(n-1); this represents a set of n k-cycles, each repeating the same d elements (parts), but starting in different places.
T(n,k) = Sum_{P a partition of n} SP(P) * Product_( (sigma_{i-1}(k))^(P(i)-1) ), where SP is A036040 or A080575, and P(i) is the number of parts in P of size i.
T(n,k) = Sum_{j=0..n-1} A036073(n,j)*k^(n-1-j). - Andrey Zabolotskiy, Oct 22 2017

Offset set to 0 by Alois P. Heinz, Oct 29 2015

A080575 Triangle of multinomial coefficients, read by rows (version 2).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 4, 3, 6, 1, 1, 5, 10, 10, 15, 10, 1, 1, 6, 15, 15, 10, 60, 20, 15, 45, 15, 1, 1, 7, 21, 21, 35, 105, 35, 70, 105, 210, 35, 105, 105, 21, 1, 1, 8, 28, 28, 56, 168, 56, 35, 280, 210, 420, 70, 280, 280, 840, 560, 56, 105, 420, 210, 28, 1, 1, 9, 36, 36, 84, 252, 84, 126, 504, 378, 756, 126, 315, 1260, 1260, 1890, 1260, 126, 280, 2520, 840, 1260, 3780, 1260, 84, 945, 1260, 378, 36, 1, 1, 10, 45, 45, 120, 360, 120, 210, 840, 630, 1260, 210

Offset: 1

Wouter Meeussen, Mar 23 2003

This is different from A036040 and A178867.
T[n,m] = count of set partitions of n with block lengths given by the m-th partition of n in the canonical ordering.
From Tilman Neumann, Oct 05 2008: (Start)
These are also the coefficients occurring in complete Bell polynomials, Faa di Bruno's formula (in its simplest form) and computation of moments from cumulants.
Though the Bell polynomials seem quite unwieldy, they can be computed easily as the determinant of an n-dimensional square matrix. (see e.g. [Coffey] and program below)
The complete Bell polynomial of the first n primes gives A007446. (End)
The difference with A036040 and A178867 lies in the ordering of the monomials. This sequence uses lexicographic ordering, while in A036040 the total order (power) of the monomials prevails (Abramowitz-Stegun style): e.g., in row 6 we have ...+ 15*x[3]*x[5] + 15*x[3]*x[6]^2 + 10*x[4]^2 +...; in A036040 the coefficient of x[3]*x[6]^2 would come after that of x[4]^2 because the total order is higher, here it comes before in view of the lexicographic order. - M. F. Hasler, Jul 12 2015

			For n=4 the 5 integer partitions in canonical ordering with corresponding set partitions and counts are:
   [4]       -> #{1234} = 1
   [3,1]     -> #{123/4, 124/3, 134/2, 1/234} = 4
   [2,2]     -> #{12/34, 13/24, 14/23} = 3
   [2,1,1]   -> #{12/3/4, 13/2/4, 1/23/4, 14/2/3, 1/24/3, 1/2/34} = 6
   [1,1,1,1] -> #{1/2/3/4} = 1
Thus row 4 is [1, 4, 3, 6, 1].
Triangle begins:
1;
1, 1;
1, 3,  1;
1, 4,  3,  6,  1;
1, 5, 10, 10, 15,  10,  1;
1, 6, 15, 15, 10,  60, 20, 15,  45,  15,  1;
1, 7, 21, 21, 35, 105, 35, 70, 105, 210, 35, 105, 105, 21, 1;
...
Row 4 represents 1*k(4)+4*k(3)*k(1)+3*k(2)^2+6*k(2)*k(1)^2+1*k(1)^4 and T(4,4)=6 since there are six ways of partitioning four labeled items into one part with two items and two parts each with one item.

See A036040 for the column labeled "M_3" in Abramowitz and Stegun, Handbook, p. 831.

Alois P. Heinz, Rows n = 1..26, flattened
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Berg, Kimmo Complexity of solution structures in nonlinear pricing, Ann. Oper. Res. 206, 23-37 (2013).
R. J. Cano, Sequencer program.
Mark W. Coffey, A Set of Identities for a Class of Alternating Binomial Sums Arising in Computing Applications, arXiv:math-ph/0608049, 2006.
Wikipedia, Cumulant.
Wikipedia, Bell polynomials

See A036040 for another version. Cf. A036036-A036039.
Row sums are A000110.
Row lengths are A000041.
Cf. A007446. - Tilman Neumann, Oct 05 2008
Cf. A178866 and A178867 (version 3). - Johannes W. Meijer, Jun 21 2010
Maximum value in row n gives A102356(n).

runs[li:{__Integer}] := ((Length/@ Split[ # ]))&[Sort@ li]; Table[Apply[Multinomial, IntegerPartitions[w], {1}]/Apply[Times, (runs/@ IntegerPartitions[w])!, {1}], {w, 6}]
(* Second program: *)
completeBellMatrix[x_, n_] := Module[{M, i, j}, M[, ] = 0; For[i=1, i <= n-1 , i++, M[i, i+1] = -1]; For[i=1, i <= n , i++, For[j=1, j <= i, j++, M[i, j] = Binomial[i-1, j-1]*x[i-j+1]]]; Array[M, {n, n}]]; completeBellPoly[x_, n_] := Det[completeBellMatrix[x, n]]; row[n_] := List @@ completeBellPoly[x, n] /. x[] -> 1 // Reverse; Table[row[n], {n, 1, 10}] // Flatten (* _Jean-François Alcover, Aug 31 2016, after Tilman Neumann *)
B[0] = 1;
B[n_] := B[n] = Sum[Binomial[n-1, k] B[n-k-1] x[k+1], {k, 0, n-1}]//Expand;
row[n_] := Reverse[List @@ B[n] /. x[_] -> 1];
Table[row[n], {n, 1, 10}] // Flatten (* Jean-François Alcover, Aug 10 2018, after Wolfdieter Lang *)

completeBellMatrix := proc(x,n) // x - vector x[1]...x[m], m>=n
local i,j,M; begin M:=matrix(n,n): // zero-initialized
for i from 1 to n-1 do M[i,i+1]:=-1: end_for:
for i from 1 to n do for j from 1 to i do
    M[i,j] := binomial(i-1,j-1)*x[i-j+1]:
end_for: end_for:
return (M): end_proc:
completeBellPoly := proc(x, n) begin
return (linalg::det(completeBellMatrix(x,n))): end_proc:
for i from 1 to 10 do print(i,completeBellPoly(x,i)): end_for:
// Tilman Neumann, Oct 05 2008

PARI
```
\\ See links.
```

A080575_poly(n,V=vector(n,i,eval(Str('x,i))))={matdet(matrix(n,n,i,j,if(j<=i,binomial(i-1,j-1)*V[n-i+j],-(j==i+1))))}
A080575_row(n)={(f(s)=if(type(s)!="t_INT",concat(apply(f,select(t->t,Vec(s)))),s))(A080575_poly(n))} \\ M. F. Hasler, Jul 12 2015

A127671 Cumulant expansion numbers: Coefficients in expansion of log(1 + Sum_{k>=1} x[k]*(t^k)/k!).

Original entry on oeis.org

1, 1, -1, 1, -3, 2, 1, -4, -3, 12, -6, 1, -5, -10, 20, 30, -60, 24, 1, -6, -15, -10, 30, 120, 30, -120, -270, 360, -120, 1, -7, -21, -35, 42, 210, 140, 210, -210, -1260, -630, 840, 2520, -2520, 720, 1, -8, -28, -56, -35, 56, 336, 560, 420, 560, -336, -2520, -1680, -5040, -630, 1680, 13440, 10080, -6720

Offset: 1

Wolfdieter Lang, Jan 23 2007

Connected objects from general (disconnected) objects.
The row lengths of this array is p(n):=A000041(n) (partition numbers).
In row n the partitions of n are taken in the Abramowitz-Stegun order.
One could call the unsigned numbers |a(n,k)| M_5 (similar to M_0, M_1, M_2, M_3 and M_4 given in A111786, A036038, A036039, A036040 and A117506, resp.).
The inverse relation (disconnected from connected objects) is found in A036040.
(d/da(1))p_n[a(1),a(2),...,a(n)] = n b_(n-1)[a(1),a(2),...,a(n-1)], where p_n are the partition polynomials of the cumulant generator A127671 and b_n are the partition polynomials for A133314. - Tom Copeland, Oct 13 2012
See notes on relation to Appell sequences in a differently ordered version of this array A263634. - Tom Copeland, Sep 13 2016
Given a binomial Sheffer polynomial sequence defined by the e.g.f. exp[t * f(x)] = Sum_{n >= 0} p_n(t) * x^n/n!, the cumulants formed from these polynomials are the Taylor series coefficients of f(x) multipied by t. An example is the sequence of the Stirling polynomials of the first kind A008275 with f(x) = log(1+x), so the n-th cumulant is (-1)^(n-1) * (n-1)! * t. - Tom Copeland, Jul 25 2019
From Tom Copeland, Oct 15 2020: (Start)
With a_n = n! * b_n = (n-1)! * c_n for n  > 0, represent a function with f(0) = a_0 = b_0 = 1 as an
A) exponential generating function (e.g.f), or formal Taylor series: f(x) = e^{a.x} = 1 + Sum_{n > 0} a_n * x^n/n!
B) ordinary generating function (o.g.f.), or formal power series: f(x)  = 1/(1-b.x) = 1 + Sum_{n > 0}  b_n * x^n
C) logarithmic generating function (l.g.f): f(x) = 1 - log(1 - c.x) = 1 + Sum_{n > 0}  c_n * x^n /n.
Expansions of log(f(x)) are given in
I) A127671 and A263634 for the e.g.f: log[ e^{a.*x} ] =  e^{L.(a_1,a_2,...)x} = Sum_{n > 0} L_n(a_1,...,a_n) * x^n/n!, the logarithmic polynomials, cumulant expansion polynomials
II) A263916 for the o.g.f.: log[ 1/(1-b.x) ] =  log[ 1 - F.(b_1,b_2,...)x ] = -Sum_{n > 0} F_n(b_1,...,b_n) * x^n/n, the Faber polynomials.
Expansions of exp(f(x)-1) are given in
III) A036040 for an e.g.f: exp[ e^{a.x} - 1 ] = e^{BELL.(a_1,...)x}, the Bell/Touchard/exponential partition polynomials, a.k.a. the Stirling partition polynomials of the second kind
IV) A130561 for an o.g.f.: exp[ b.x/(1-b.x) ] = e^{LAH.(b.,...)x}, the Lah partition polynomials
V) A036039 for an l.g.f.: exp[ -log(1-c.x) ] =  e^{CIP.(c_1,...)x}, the cycle index polynomials of the symmetric groups S_n, a.k.a. the Stirling partition polynomials of the first kind.
Since exp and log are a compositional inverse pair, one can extract the indeterminates of the log set of partition polynomials from the exp set and vice versa. For a discussion of the relations among these polynomials and the combinatorics of connected and disconnected graphs/maps, see Novak and LaCroix on classical moments and cumulants and the two books on statistical mechanics referenced in A036040. (End)
Ignoring signs, these polynomials appear in Schröder in the set of equations (II) on p. 343 and in Stewart's translation on p. 31. - Tom Copeland, Aug 25 2021

			Row n=3: [1,-3,2] stands for the polynomial 1*x[3] - 3*x[1]*x[2] + 2*x[1]^3 (the Abramowitz-Stegun order of the p(3)=3 partitions of n=3 is [3],[1,2],[1^3]).

C. Itzykson and J.-M. Drouffe, Statistical field theory, vol. 2, p. 413, eq.(13), Cambridge University Press, (1989).

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, pp. 831-2.
Tom Copeland, The creation / raising operators for Appell sequences.
Wolfdieter Lang, First 10 rows of cumulant numbers and polynomials
E. Schröder, Ueber unendlich viele Algorithmen zur Auflösung der Gleichungen, Mathematische Annalen vol. 2, 317-365, 1870.
G. Stewart, On infinitely many algorithms for solving equations, 1993, (translation into English of Schröder's paper above).

Cf. A133314, A263916, A263634.
Cf. A008275.
Cf. A036039, A036040, A130561.

E.g.f. for multivariate row polynomials A(t) := log(1 + Sum_{k>=1} x[k]*(t^k)/k!).
Row n polynomial p_n(x[1],...,x[n]) = [(t^n)/n!]A(t).
a(n,m) = A264753(n, m) * M_3(n,m) with M_3(n,m) = A036040(n,m) (Abramowitz-Stegun M_3 numbers). - corrected by Johannes W. Meijer, Jul 12 2016
p_n(x[1],...,x[n]) = -(n-1)!*F(n,x[1],x[2]/2!,..,x[n]/n!) in terms of the Faber polynomials F(n,b1,..,bn) of A263916. - Tom Copeland, Nov 17 2015
With D = d/dz and M(0) = 1, the differential operator R = z + d(log(M(D))/dD = z + d(log(1 + x[1] D + x[2] D^2/2! + ...))/dD = z + p.*exp(p.D) = z + Sum_{n>=0} p_(n+1)(x[1],..,x[n]) D^n/n! is the raising operator for the Appell sequence A_n(z) = (z + x[.])^n = Sum_{k=0..n} binomial(n,k) x[n-k] z^k with the e.g.f. M(t) e^(zt), i.e., R A_n(z) = A_(n+1)(z) and dA_n(z)/dz = n A_(n-1)(z). The operator Q = z - p.*exp(p.D) generates the Appell sequence with e.g.f. e^(zt) / M(t). - Tom Copeland, Nov 19 2015

A156289 Triangle read by rows: T(n,k) is the number of end rhyme patterns of a poem of an even number of lines (2n) with 1<=k<=n evenly rhymed sounds.

Original entry on oeis.org

1, 1, 3, 1, 15, 15, 1, 63, 210, 105, 1, 255, 2205, 3150, 945, 1, 1023, 21120, 65835, 51975, 10395, 1, 4095, 195195, 1201200, 1891890, 945945, 135135, 1, 16383, 1777230, 20585565, 58108050, 54864810, 18918900, 2027025, 1, 65535, 16076985

Offset: 1

Hartmut F. W. Hoft, Feb 07 2009

T(n,k) is the number of partitions of a set of size 2*n into k blocks of even size [Comtet]. For partitions into odd sized blocks see A136630.
See A241171 for the triangle of ordered set partitions of the set {1,2,...,2*n} into k even sized blocks. - Peter Bala, Aug 20 2014
This triangle T(n,k) gives the sum over the M_3 multinomials A036040 for the partitions of 2*n with k even parts, for 1 <= k <= n. See the triangle A257490 with sums over the entries with k parts, and the Hartmut F. W. Hoft program. - Wolfdieter Lang, May 13 2015

			The triangle begins
  n\k|..1.....2......3......4......5......6
  =========================================
  .1.|..1
  .2.|..1.....3
  .3.|..1....15.....15
  .4.|..1....63....210....105
  .5.|..1...255...2205...3150....945
  .6.|..1..1023..21120..65835..51975..10395
  ..
T(3,3) = 15. The 15 partitions of the set [6] into three even blocks are:
  (12)(34)(56), (12)(35)(46), (12)(36)(45),
  (13)(24)(56), (13)(25)(46), (13)(26)(45),
  (14)(23)(56), (14)(25)(36), (14)(26)(35),
  (15)(23)(46), (15)(24)(36), (15)(26)(34),
  (16)(23)(45), (16)(24)(35), (16)(25)(34).
Examples of recurrence relation
 T(4,3) = 5*T(3,2) + 9*T(3,3) = 5*15 + 9*15 = 210;
 T(6,5) = 9*T(5,4) + 25*T(5,5) = 9*3150 + 25*945 = 51975.
 T(4,2) = 28 + 35 = 63 (M_3 multinomials A036040 for partitions of 8 with 3 even parts, namely (2,6) and (4^2)). - _Wolfdieter Lang_, May 13 2015

L. Comtet, Analyse Combinatoire, Presses Univ. de France, 1970, Vol. II, pages 61-62.
L. Comtet, Advanced Combinatorics, Reidel, 1974, pages 225-226.

Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows 1 <= n <= 150, flattened)
Richard Olu Awonusika, On Jacobi polynomials P_k(alpha, beta) and coefficients c_j^L(alpha, beta) (k >= 0, L = 5,6; 1 <= j <= L; alpha, beta > -1), The Journal of Analysis (2020).
Thomas Browning, Counting Parabolic Double Cosets in Symmetric Groups, arXiv:2010.13256 [math.CO], 2020.
J. Riordan, Letter, Jul 06 1978

Diagonal T(n, n) is A001147, subdiagonal T(n+1, n) is A001880.
2nd column variant T(n, 2)/3, for 2<=n, is A002450.
3rd column variant T(n, 3)/15, for 3<=n, is A002451.
Sum of the n-th row is A005046.
Cf. A241171, A257468, A257490, A096162.

T := proc(n,k) option remember; `if`(k = 0 and n = 0, 1, `if`(n < 0, 0,
(2*k-1)*T(n-1, k-1) + k^2*T(n-1, k))) end:
for n from 1 to 8 do seq(T(n,k), k=1..n) od; # Peter Luschny, Sep 04 2017

T[n_,k_] := Which[n < k, 0, n == 1, 1, True, 2/Factorial2[2 k] Sum[(-1)^(k + j) Binomial[2 k, k + j] j^(2 n), {j, 1, k}]]
(* alternate computation with function triangle[] defined in A257490 *)
a[n_]:=Map[Apply[Plus,#]&,triangle[n],{2}]
(* Hartmut F. W. Hoft, Apr 26 2015 *)

Recursion: T(n,1)=1 for 1<=n; T(n,k)=0 for 1<=n

Generating function for the k-th column of the triangle T(i+k,k):

G(k,x) = Sum_{i>=0} T(i+k,k)*x^i = Product_{j=1..k} (2*j-1)/(1-j^2*x).

Closed form expression: T(n,k) = (2/(k!*2^k))*Sum_{j=1..k} (-1)^(k-j)*binomial(2*k,k-j)*j^(2*n).

From Peter Bala, Feb 21 2011: (Start)

GENERATING FUNCTION

E.g.f. (including a constant 1):

(1)... F(x,z) = exp(x*(cosh(z)-1))

= Sum_{n>=0} R(n,x)*z^(2*n)/(2*n)!

= 1 + x*z^2/2! + (x + 3*x^2)*z^4/4! + (x + 15*x^2 + 15*x^3)*z^6/6! + ....

ROW POLYNOMIALS

The row polynomials R(n,x) begin

... R(1,x) = x

... R(2,x) = x + 3*x^2

... R(3,x) = x + 15*x^2 + 15*x^3.

The egf F(x,z) satisfies the partial differential equation

(2)... d^2/dz^2(F) = x*F + x*(2*x+1)*F' + x^2*F'',

where ' denotes differentiation with respect to x. Hence the row polynomials satisfy the recurrence relation

(3)... R(n+1,x) = x*{R(n,x) + (2*x+1)*R'(n,x) + x*R''(n,x)}

with R(0,x) = 1. The recurrence relation for T(n,k) given above follows from this.

(4)... T(n,k) = (2*k-1)!!*A036969(n,k).

(End)

A049019 Irregular triangle read by rows: Row n gives numbers of preferential arrangements (onto functions) of n objects that are associated with the partition of n, taken in Abramowitz and Stegun order.

Original entry on oeis.org

1, 1, 2, 1, 6, 6, 1, 8, 6, 36, 24, 1, 10, 20, 60, 90, 240, 120, 1, 12, 30, 20, 90, 360, 90, 480, 1080, 1800, 720, 1, 14, 42, 70, 126, 630, 420, 630, 840, 5040, 2520, 4200, 12600, 15120, 5040, 1, 16, 56, 112, 70, 168, 1008, 1680, 1260, 1680, 1344, 10080, 6720

Offset: 1

Alford Arnold

This is a refinement of A019538 with row sums in A000670.
From Tom Copeland, Sep 29 2008: (Start)
This array is related to the reciprocal of an e.g.f. as sketched in A133314. For example, the coefficient of the fourth-order term in the Taylor series expansion of 1/(a(0) + a(1) x + a(2) x^2/2! + a(3) x^3/3! + ...) is a(0)^(-5) * {24 a(1)^4 - 36 a(1)^2 a(2) a(0) + [8 a(1) a(3) + 6 a(2)^2] a(0)^2 - a(4) a(0)^3}.
The unsigned coefficients characterize the P3 permutohedron depicted on page 10 in the Loday link with 24 vertices (0-D faces), 36 edges (1-D faces), 6 squares (2-D faces), 8 hexagons (2-D faces) and 1 3-D permutohedron. Summing coefficients over like dimensions gives A019538 and A090582. Compare to A133437 for the associahedron.
Given the n X n lower triangular matrix M = [ binomial(j,k) u(j-k) ], the first column of the inverse matrix M^(-1) contains the (n-1) rows of A049019 as the coefficients of the multinomials formed from the u(j). M^(-1) can be computed as (1/u(0)){I - [I- M/u(0)]^n} / {I - [I- M/u(0)]} = - u(0)^(-n) {sum(j=1 to n)(-1)^j bin(n,j) u(0)^(n-j) M^(j-1)} where I is the identity matrix.
Another method for computing the coefficients and partitions up to (n-1) rows is to use (1-x^n)/ (1-x) = 1+x^2+x^3+ ... + x^(n-1) with x replaced either by [I- M/a(0)] or [1- g(x)/a(0)] with the n X n matrix M = [bin(j,k) a(j-k)] and g(x)= a(0) + a(1)x + a(2)x^2/2! + ... + a(n) x^n/n!. The first n terms (rows of the first column) of the resulting series (matrix) divided by a(0) contain the (n-1) rows of signed coefficients and associated partitions for A049019.
To obtain unsigned coefficients, change a(j) to -a(j) for j>0. A133314 contains other matrices and recursion formulas that could be used. The Faa di Bruno formula gives the coefficients as n! [e(1)+e(2)+...+e(n)]! / [1!^e(1) e(1)! 2!^e(2) e(2)!... n!^e(n) e(n)! ] for the partition of form [a(1)^e(1)...a(n)^e(n)] with [e(1)+2e(2)+...+ n e(n)] = n (see Abramowitz and Stegun pages 823 and 831) in agreement with Arnold's formula. (End)

			Irregular triangle starts (note the grouping by ';' when comparing with A019538):
[1] 1;
[2] 1;  2;
[3] 1;  6;  6;
[4] 1;  8,  6; 36;  24;
[5] 1; 10, 20; 60,  90; 240; 120;
[6] 1; 12, 30, 20;  90, 360,  90; 480, 1080; 1800; 720;
[7] 1; 14, 42, 70; 126, 630, 420, 630;  840, 5040, 2520; 4200, 12600; 15120; 5040;
.
a(17) = 240 because we can write
A048996(17)*A036038(17) = 4*60 = A036040(17)*A036043(17)! = 10*24.
As in A133314, 1/exp[u(.)*x] = u(0)^(-1) [ 1 ] + u(0)^(-2) [ -u(1) ] x + u(0)^(-3) [ -u(0)u(2) + 2 u(1)^2 ] x^2/2! + u(0)^(-4) [ -u(0)^2 u(3) + 6 u(0)u(1)u(2) - 6 u(1)^3 ] x^3/3! + u(0)^(-5) [ -u(0)^3 u(4) + 8 u(0)^2 u(1)u(3) + 6 u(0)^2 u(2)^2 - 36 u(0)u(1)^2 u(2) + 24 u(1)^4 ] x^4/4! + ... . These are essentially refined face polynomials for permutohedra: empty set + point + line segment + hexagon + 3-D- permutohedron + ... . - _Tom Copeland_, Oct 04 2008

Peter Luschny, Rows n = 1..36, flattened
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972.
N. Arkani-Hamed, Y. Bai, S. He, and G. Yan, Scattering forms and the positive geometry of kinematics, color, and the worldsheet , arXiv:1711.09102 [hep-th], 2017.
Tom Copeland, Bijective mapping between face polytopes of permutohedra and partitions of integers, Math StackExchange question, 2016.
S. Forcey, The Hedra Zoo.
X. Gao, S. He, and Y. Zhan, Labelled tree graphs, Feynman diagrams and disk integrals , arXiv:1708.08701 [hep-th], 2017.
J. Loday, The Multiple Facets of the Associahedron
V. Pilaud, The Associahedron and its Friends, presentation for Seminaire Lotharingien de Combinatoire, April 4 - 6, 2016.
A. Postnikov, Positive Grassmannian and Polyhedral Subdivisions, arXiv:1806.05307 [math.CO], (cf. p. 17), 2018.

Cf. A000041, A000670, A019538, A036038, A036040, A036043, A048996, A133314.

def A049019(n):
    if n == 0: return [1]
    P = lambda k: Partitions(n, min_length=k, max_length=k)
    Q = (p.to_list() for k in (1..n) for p in P(k))
    return [factorial(len(p))*SetPartitions(sum(p), p).cardinality() for p in Q]
for n in (1..7): print(A049019(n)) # Peter Luschny, Aug 30 2019

a(n) = A048996(n) * A036038(n);
a(n) = A036040(n) * factorial(A036043(n)).
A lowering operator for the unsigned multinomials in the brackets in the example is [d/du(1) 1/POP] where u(1) is treated as a continuous variable and POP is an operator that pulls off the number of parts of a partition ignoring u(0), e.g., [d/du(1) 1/POP][ u(0)u(2) + 2 u(1)^2 ] = (1/2) 2*2 u(1) = 2*u(1), analogous to the prototypical delta operator (d/dz) z^n = n z^(n-1). - Tom Copeland, Oct 04 2008
From the matrix formulation with M_m,k = 1/(m-k)!; g(x) = exp[ u(.) x]; an orthonormal vector basis x_1, ..., x_n and En(x^k) = x_k for k <= n and zero otherwise, for j=0 to n-1 the j-th signed row multinomial is given by the wedge product of x_1 with the wedge product (-1)^j * j! * u(0)^(-n) * Wedge{ En[x g(x), x^2 g(x), ..., x^(j) g(x), ~, x^(j+2) g(x), ..., x^n g(x)] } where Wedge{a,b,c} = a v b v c (the usual wedge symbol is inverted here to prevent confusion with the power notation, see Mathworld) and the (j+1)-th element is omitted from the product. Tom Copeland, Oct 06 2008 [Changed an x^n to x^(n-1) and "inner product of x_1" to "wedge". - Tom Copeland, Feb 03 2010]

Partitions for 7 and 8 from Tom Copeland, Oct 02 2008

Definition edited by N. J. A. Sloane, Nov 06 2023

A007446 Exponentiation of e.g.f. for primes.

Original entry on oeis.org

1, 2, 7, 31, 162, 973, 6539, 48410, 390097, 3389877, 31534538, 312151125, 3271508959, 36149187780, 419604275375, 5100408982825, 64743452239424, 856157851884881, 11768914560546973, 167841252874889898, 2479014206472819045, 37860543940437797897

Offset: 0

N. J. A. Sloane

From Tilman Neumann, Oct 05 2008: (Start)
a(n) is also given by
- substituting the primes (A000040) into (the simplest) Faa di Bruno's formula, or
- the complete Bell polynomial of the first n prime arguments, or
- computing n-th moments from the first n primes as cumulants
The examples show that the coefficients of the prime power products are just A036040/A080575 (these are just rearrangements of the same coefficients). Moreover, the prime products of the additional terms span the whole space of natural numbers, thus what we see here is a reordering of the natural numbers! (End)

			From _Tilman Neumann_, Oct 05 2008: (Start)
Let p_i denote the i-th prime A000040(i). Then
a(1)=2 = 1*p_1
a(2)=7 = 1*p_2 + 1*p_1^2
a(3)=31 = 1*p_3 + 3*p_2*p_1 + 1*p_1^3
a(4)=162= 1*p_4 + 4*p_3*p_1 + 3*p_2^2 + 6*p_2*p_1^2 + 1*p_1^4
a(5)=973= 1*p_5 + 5*p_4*p_1 + 10*p_3*p_2 + 10*p_3*p_1^2 + 15*p_2^2*p_1 + 10*p_2*p_1^3 + 1*p_1^5
(End)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..500

Cf. A036040, A080575. - Tilman Neumann, Oct 05 2008

a:= proc(n) option remember; `if`(n=0, 1,
      add(binomial(n-1, j-1)*ithprime(j)*a(n-j), j=1..n))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Mar 18 2015

a[n_] := a[n] = If[n==0, 1, Sum[Binomial[n-1, j-1]*Prime[j]*a[n-j], {j, 1, n}]]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 30 2015, after Alois P. Heinz *)
Table[Sum[BellY[n, k, Prime[Range[n]]], {k, 0, n}], {n, 0, 20}] (* Vladimir Reshetnikov, Nov 09 2016 *)

completeBellMatrix := proc(x,n)
// x - vector x[1]...x[m], m>=n
local i,j,M;
begin
M:=matrix(n,n): // zero-initialized
for i from 1 to n-1 do
M[i,i+1]:=-1:
end_for:
for i from 1 to n do
for j from 1 to i do
M[i,j] := binomial(i-1,j-1)*x[i-j+1]:
end_for:
end_for:
return (M):
end_proc:
completeBellPoly := proc(x, n)
begin
return (linalg::det(completeBellMatrix(x,n))):
end_proc:
x:=[2,3,5,7,11,13,17,19,23,29]:
for i from 1 to 10 do print(i,completeBellPoly(x,i)): end_for:
// Tilman Neumann, Oct 05 2008

E.g.f.: exp(Sum_{k>=1} prime(k)*x^k/k!). - Ilya Gutkovskiy, Nov 26 2017

A115131 Waring numbers for power sums functions in terms of elementary symmetric functions; irregular triangle T(n,k), read by rows, for n >= 1 and 1 <= k <= A000041(n).

Original entry on oeis.org

1, -2, 1, 3, -3, 1, -4, 4, 2, -4, 1, 5, -5, -5, 5, 5, -5, 1, -6, 6, 6, 3, -6, -12, -2, 6, 9, -6, 1, 7, -7, -7, -7, 7, 14, 7, 7, -7, -21, -7, 7, 14, -7, 1, -8, 8, 8, 8, 4, -8, -16, -16, -8, -8, 8, 24, 12, 24, 2, -8, -32, -16, 8, 20, -8, 1, 9, -9, -9, -9, -9, 9, 18, 18, 9, 9, 18, 3, -9, -27, -27, -27, -27, -9, 9, 36, 18, 54, 9, -9, -45, -30, 9, 27, -9, 1

Offset: 1

Wolfdieter Lang, Jan 13 2006

N*t^{(N)}n(sigma_1, ..., sigma_N):= sum((x_k)^n, k=1..N) with the elementary symmetric function sigma_k (superscript (N) omitted) in terms of the indeterminates x_1,...,x_N, is an N-variable generalization of Chebyshev's polynomials C_n((sigma_1)/2) = t^{(N=2)}_n(sigma_1, sigma_2 = 1). In general, C_n^{(N)}(sigma_1, ..., sigma{N-1}) := t^{(N)}n(sigma_1, ..., sigma{N-1}, sigma_N:=1). If n > N, one puts sigma_{N+1} = 0, ..., sigma_n = 0.
The sequence of row lengths of this array is A000041(n) (partition numbers).
In row n, this triangular array uses partitions of n listed in the Abramowitz-Stegun order (compare with the M_0, M_1, M_2 and M_3 numbers given in A048996 = |A111786|, A036038, A036039 and A036040, resp.).
Row sums give (-1)^(n-1). Unsigned row sums give A000225(n)= 2^n - 1.
N*t^{(N)}n(sigma_1, ..., sigma_N) gives the sum of the n-th power of the indeterminates x_1, ... , x_N in terms of the elementary symmetric functions of these indeterminates. The coefficient T(n, k) of this partition array corresponds to the k-th partition of n in the Abramowitz-Stegun order, and it multiplies the product of sigma_j functions encoded by this partition. See the example for n = 4 below. - _Wolfdieter Lang, Mar 09 2015

			First few rows of triangle T(n,k) are as follows (see the link for rows 1..10):
   1;
  -2,  1;
   3, -3,  1;
  -4,  4,  2, -4, 1;
   5, -5, -5,  5, 5, -5, 1;
  ...
n=4: N*t^{(N)}_4 = -4*(sigma_4)^1 + 4*(sigma_1)*(sigma_3) + 2*(sigma_2)^2 -4*(sigma_1)^2*(sigma_2) + 1*(sigma_1)^4.
  (For 2 <= N < 4, one puts sigma_{N+1} = 0 = ... = sigma_4 = 0.) This becomes Sum_{k = 1..N} (x_k)^4 if the sigma functions are written in terms of the variables x_1, x_2, ..., x_N. E.g., for N=2: 0 + 0 + 2*(x_1*x_2)^2 -4*(x_1 + x_2)^2*(x_1*x_2) + 1*(x_1 + x_2)^4 = (x_1)^4 + (x_2)^4.

P. A. MacMahon, Combinatory Analysis, 2 vols., Chelsea, NY, 1960, see p. 5 (with a_k -> sigma_k).

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972; see pp. 831-832. [alternative scanned copy].
Wolfdieter Lang, On sums of powers of zeros of polynomials, J. Comp. Appl. Math. 89 (1998) 237-256; see Theorem 1.
Wolfdieter Lang, First 10 rows of the array.
R. Lidl, Tschebyscheffpolynome in mehreren Variablen, J. reine u. angew. Math. 273 (1975), 178-198.
R. Lidl, Tschebyscheffpolynome in mehreren Variablen, J. reine u. angew. Math. 273 (1975), 178-198.
R. Lidl and Ch. Wells, Chebyshev polynomials in several variables, J. reine u. angew. Math. 255 (1972), 104-111.
R. Lidl and Ch. Wells, Chebyshev polynomials in several variables, J. reine u. angew. Math. 255 (1972), 104-111.
P. A. MacMahon, Combinatory analysis (2 vols.), Chelsea, NY, 1960; see p. 5 (with a_k -> sigma_k).

Cf. A000041, A000225, A036038, A036039, A036040, A048996, A111786.
Cf. A210258 (in another ordering of partitions), A132460 (N=2), A325477 (N=3),
A324602 (N=4).

T(n,k) = (n/m(n,k))*A111786(n,k) for the k-th partition of n with m(n,k) parts in the Abramowitz-Stegun order for n >= 1 and k = 1..p(n), where p(n) := A000041(n).

Explicitly: T(n,k) = (-1)^(n + m(n,k)) * n * (m(n,k) - 1)!/(Product_{j = 1..n} e(k,j)!), where m(n,k):= Sum_{j = 1..n} e(k,j), with [1^e(k, 1), 2^e(k,2), ..., n^e(k,n)] being the k-th partition of n in the mentioned order. For m(n,k), see A036043.

Various sections edited by Petros Hadjicostas, Dec 14 2019

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