A112007 -id:A112007 - cp's OEIS frontend

A111999 T(n, k) = [x^k] (-1)^nSum_{k=0..n} E2(n, n-k)(1+x)^(n-k) where E2(n, k) are the second-order Eulerian numbers. Triangle read by rows, T(n, k) for n >= 1 and 0 <= k <= n.

-1, 3, 2, -15, -20, -6, 105, 210, 130, 24, -945, -2520, -2380, -924, -120, 10395, 34650, 44100, 26432, 7308, 720, -135135, -540540, -866250, -705320, -303660, -64224, -5040, 2027025, 9459450, 18288270, 18858840, 11098780, 3678840, 623376, 40320, -34459425, -183783600, -416215800

Offset: 1

Wolfdieter Lang, Sep 12 2005

Previous name was: A triangle that converts certain binomials into triangle A008276 (diagonals of signed Stirling1 triangle A008275).
Stirling1(n,n-m) = A008275(n,n-m) = Sum_{k=0..m-1}a(m,k)*binomial(n,2*m-k).
The unsigned column sequences start with A001147, A000906 = 2*A000457, 2*|A112000|, 4*|A112001|.
The general results on the convolution of the refined partition polynomials of A133932, with u_1 = 1 and u_n = -t otherwise, can be applied here to obtain results of convolutions of these unsigned polynomials. - Tom Copeland, Sep 20 2016

			Triangle starts:
  [1]      -1;
  [2]       3,       2;
  [3]     -15,     -20,       -6;
  [4]     105,     210,      130,       24;
  [5]    -945,   -2520,    -2380,     -924,     -120;
  [6]   10395,   34650,    44100,    26432,     7308,     720;
  [7] -135135, -540540,  -866250,  -705320,  -303660,  -64224,  -5040;
  [8] 2027025, 9459450, 18288270, 18858840, 11098780, 3678840, 623376, 40320.

Charles Jordan, Calculus of Finite Differences, Chelsea 1965, p. 152. Table C_{m, nu}.

Peter Bala, Diagonals of triangles with generating function exp(t*F(x)).
Steve Butler and Pavel Karasik, A note on nested sums, J. Int. Seq. 13 (2010), 10.4.4, page 4.
Tom Copeland, Generators, Inversion, and Matrix, Binomial, and Integral Transforms
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. 6.
EqWorld, Integral Transforms
David J. Jeffrey, G. A. Kalugin, N. Murdoch, Lagrange inversion and Lambert W, 17th Int'l Symp. Symb. Numer. Algor. Sci. Comp. (SYNASC 2015).
Wolfdieter Lang, First 10 rows.
Lajos Takács, On the number of distinct forests, SIAM J. Discrete Math., 3 (1990), 574-581. Table 3 gives an unsigned version of the triangle.

Row sums give A032188(m+1)*(-1)^m, m>=1. Unsigned row sums give A032188(m+1), m>=1.
Cf. A008517 (second-order Eulerian triangle) for a similar formula for |Stirling1(n, n-m)|.
Cf. A000457, A000906, A001147, A008275, A008276, A008306, A112001, A133932,

CoeffList := p -> op(PolynomialTools:-CoefficientList(p, x)):
E2 := (n, k) -> combinat[eulerian2](n, k):
poly := n -> (-1)^n*add(E2(n, n-k)*(1+x)^(n-k), k = 0..n):
seq(CoeffList(poly(n)), n = 1..8); # Peter Luschny, Feb 05 2021

a[m_, k_] := a[m, k] = Which[m < k + 1, 0, And[m == 1, k == 0], -1, k == -1, 0, True, -(2 m - k - 1)*(a[m - 1, k] + a[m - 1, k - 1])]; Table[a[m, k], {m, 9}, {k, 0, m - 1}] // Flatten (* Michael De Vlieger, Sep 23 2016 *)

a(m, k)=0 if m

From Tom Copeland, May 05 2010 (updated Sep 12 2011): (Start)

The integral from 0 to infinity w.r.t. w of

exp[-w(u+1)] (1+u*z*w)^(1/z) gives a power series, f(u,z), in z for reversed row polynomials in u of A111999, related to an Euler transform of diagonals of A008275.

Let g(u,x) be obtained from f(u,z) by replacing z^n with x^(n+1)/(n+1)!;

g(u,x)= x - u^2 x^2/2! + (2 u^3 + 3 u^4) x^3/3! - (6 u^4 + 20 u^5 + 15 u^6) x^4/4! + ... , an e.g.f. associated to f(u,z).

Then g^(-1)(u,x)=(1+u)*x - log(1+u*x) is the comp. inverse of g(u,x) in x, and, consequently, A133932 is a refinement of A111999.

With h(u,x)= 1/(dg^(-1)/dx)= (1+u*x)/(1+(1+u)*u*x),

g(u,x)=exp[x*h(u,t)d/dt] t, evaluated at t=0. Also, dg(u,x)/dx = h(u,g(u,x)). (End)

From Tom Copeland, May 06 2010: (Start)

For m,k>0, a(m,k) = Sum(j=2 to 2m-k+1): (-1)^(2m-k+1+j) C(2m-k+1,j) St1d(j,m),

where C(n,j) is the binomial coefficient and St1d(j,m) is the (j-m)-th element of the m-th subdiagonal of A008275 for (j-m)>0 and is 0 otherwise,

e.g., St1d(1,1) = 0, St1d(2,1) = -1, St1d(3,1) = -3, St1d(4,1) = -6. (End)

From Tom Copeland, Sep 03 2011 (updated Sep 12 2011): (Start)

The integral from 0 to infinity w.r.t. w of

exp[-w*(u+1)/u] (1+u*z*w)^(1/(u^2*z)) gives a power series, F(u,z), in z for the row polynomials in u of A111999.

Let G(u,x) be obtained from F(u,z) by replacing z^n with x^(n+1)/(n+1)!;

G(u,x) = x - x^2/2! + (3 + 2 u) x^3/3! - (15 + 20 u + 6 u^2) x^4/4! + ... , an e.g.f. for A111999 associated to F(u,z).

G^(-1)(u,x) = ((1+u)*u*x - log(1+u*x))/u^2 is the comp. inverse of G(u,x) in x.

With H(u,x) = 1/(dG^(-1)/dx) = (1+u*x)/(1+(1+u)*x),

G(u,x) = exp[x*H(u,t)d/dt] t, evaluated at t=0. Also, dG(u,x)/dx = H(u,G(u,x)). (End)

From Tom Copeland, Sep 16 2011: (Start)

f(u,z) and F(u,z) are expressible in terms of the incomplete gamma function Γ(v,p)(see Laplace Transforms for Power-law Functions at EqWorld):

With K(p,s) = p^(-s-1) exp(p) Γ(s+1,p),

f(u,z) = K(p,s)/(u*z) with p=(u+1)/(u*z) and s=1/z , and

F(u,z) = K(p,s)/(u*z) with p=(u+1)/(u^2*z) and s=1/(u^2*z). (End)

Diagonals of A008306 are reversed rows of A111999 (see P. Bala). - Tom Copeland, May 08 2012

New name from Peter Luschny, Feb 05 2021

A112486 Coefficient triangle for polynomials used for e.g.f.s for unsigned Stirling1 diagonals.

Original entry on oeis.org

1, 1, 1, 2, 5, 3, 6, 26, 35, 15, 24, 154, 340, 315, 105, 120, 1044, 3304, 4900, 3465, 945, 720, 8028, 33740, 70532, 78750, 45045, 10395, 5040, 69264, 367884, 1008980, 1571570, 1406790, 675675, 135135, 40320, 663696, 4302216, 14777620, 29957620

Offset: 0

Wolfdieter Lang, Sep 12 2005

The k-th diagonal of |A008275| appears as the k-th column in |A008276| with k-1 leading zeros.
The recurrence, given below, is derived from (d/dx)g1(k,x) - g1(k,x)= x*(d/dx)g1(k-1,x) + g1(k-1,x), k >= 1, with input g(-1,x):=0 and initial condition g1(k,0)=1, k >= 0. This differential recurrence for the e.g.f. g1(k,x) follows from the one for unsigned Stirling1 numbers.
The column sequences start with A000142 (factorials), A001705, A112487- A112491, for m=0,...,5.
The main diagonal gives (2*k-1)!! = A001147(k), k >= 1.
This computation was inspired by the Bender article (see links), where the Stirling polynomials are discussed.
The e.g.f. for the k-th diagonal, k >= 1, of the unsigned Stirling1 triangle |A008275| with k-1 leading zeros is g1(k-1,x) = exp(x)*Sum_{m=0..k-1} a(k,m)*(x^(k-1+m))/(k-1+m)!.
a(k,n) = number of lists with entries from [n] such that (i) each element of [n] occurs at least once and at most twice, (ii) for each i that occurs twice, all entries between the two occurrences of i are > i, and (iii) exactly k elements of [n] occur twice. Example: a(1,2)=5 counts 112, 121, 122, 211, 221, and a(2,2)=3 counts 1122,1221,2211. - David Callan, Nov 21 2011

			Triangle begins:
    1;
    1,    1;
    2,    5,     3;
    6,   26,    35,    15;
   24,  154,   340,   315,   105;
  120, 1044,  3304,  4900,  3465,   945;
  720, 8028, 33740, 70532, 78750, 45045, 10395;
k=3 column of |A008276| is [0,0,2,11,35,85,175,...] (see A000914), its e.g.f. exp(x)*(2*x^2/2! + 5* x^3/3! + 3*x^4/4!).

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Roland Bacher, Counting Packings of Generic Subsets in Finite Groups, Electr. J. Combinatorics, 19 (2012), #P7. - From _N. J. A. Sloane_, Feb 06 2013
C. M. Bender, D. C. Brody and B. K. Meister, Bernoulli-like polynomials associated with Stirling Numbers, arXiv:math-ph/0509008 [math-ph], 2005.
Wolfdieter Lang, First 10 rows.

Cf. A112007 (triangle for o.g.f.s for unsigned Stirling1 diagonals). A112487 (row sums).

A112486 := proc(n,k)
    if n < 0 or k<0 or  k> n then
        0 ;
    elif n = 0 then
        1 ;
    else
        (n+k)*procname(n-1,k)+(n+k-1)*procname(n-1,k-1) ;
    end if;
end proc: # R. J. Mathar, Dec 19 2013

A112486 [n_, k_] := A112486[n, k] = Which[n<0 || k<0 || k>n, 0, n == 0, 1, True, (n+k)*A112486[n-1, k]+(n+k-1)*A112486[n-1, k-1]]; Table[A112486[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 05 2014, after R. J. Mathar *)

a(k, m) = (k+m)*a(k-1, m) + (k+m-1)*a(k-1, m-1) for k >= m >= 0, a(0, 0)=1, a(k, -1):=0, a(k, m)=0 if k < m.
From Tom Copeland, Oct 05 2011: (Start)
With polynomials
P(0,t) = 0
P(1,t) = 1
P(2,t) = -(1 + t)
P(3,t) = 2 + 5 t + 3 t^2
P(4,t) = -( 6 + 26 t + 35 t^2 + 15 t^3)
P(5,t) = 24 + 154 t +340 t^2 + 315 t^3 + 105 t^4
Apparently, P(n,t) = (-1)^(n+1) PW[n,-(1+t)] where PW are the Ward polynomials A134991. If so, an e.g.f. for the polynomials is
  A(x,t) = -(x+t+1)/t - LW{-((t+1)/t) exp[-(x+t+1)/t]}, where LW(x) is a suitable branch of the Lambert W Fct. (e.g., see A135338). The comp. inverse in x (about x = 0) is B(x) = x + (t+1) [exp(x) - x - 1]. See A112487 for special case t = 1. These results are a special case of A134685 with u(x) = B(x), i.e., u_1=1 and (u_n)=(1+t) for n>0.
Let h(x,t) = 1/(dB(x)/dx) = 1/[1+(1+t)*(exp(x)-1)], an e.g.f. in x for row polynomials in t of signed A028246 , then P(n,t), is given by
(h(x,t)*d/dx)^n x, evaluated at x=0, i.e., A(x,t)=exp(x*h(u,t)*d/du) u, evaluated at u=0. Also, dA(x,t)/dx = h(A(x,t),t).
The e.g.f. A(x,t) = -v * Sum_{j>=1} D(j-1,u) (-z)^j / j! where u=-(x+t+1)/t, v=1+u, z=(1+t*v)/(t*v^2) and D(j-1,u) are the polynomials of A042977. dA/dx = -1/[t*(v-A)].(End)
A133314 applied to the derivative of A(x,t) implies (a.+b.)^n = 0^n, for (b_n)=P(n+1,t) and (a_0)=1, (a_1)=t+1, and (a_n)=t*P(n,t) otherwise. E.g., umbrally, (a.+b.)^2 = a_2*b_0 + 2 a_1*b_1 + a_0*b_2 =0. - Tom Copeland, Oct 08 2011
The row polynomials R(n,x) may be calculated using R(n,x) = 1/x^(n+1)*D^n(x), where D is the operator (x^2+x^3)*d/dx. - Peter Bala, Jul 23 2012
For n>0, Sum_{k=0..n} a(n,k)*(-1/(1+W(t)))^(n+k+1) = (t d/dt)^(n+1) W(t), where W(t) is Lambert W function. For t=-x, this gives Sum_{k>=1} k^(k+n)*x^k/k! = - Sum_{k=0..n} a(n,k)*(-1/(1+W(-x)))^(n+k+1). - Max Alekseyev, Nov 21 2019
Conjecture: row polynomials are R(n,x) = Sum_{i=0..n} Sum_{j=0..i} Sum_{k=0..j} (n+i)!*Stirling2(n+j-k,j-k)*x^k*(x+1)^(j-k)*(-1)^(n+j+k)/((n+j-k)!*(i-j)!*k!). - Mikhail Kurkov, Apr 21 2025

A002539 Eulerian numbers of the second kind: <>.

Original entry on oeis.org

1, 22, 328, 4400, 58140, 785304, 11026296, 162186912, 2507481216, 40788301824, 697929436800, 12550904017920, 236908271543040, 4687098165573120, 97049168010017280, 2099830209402931200, 47405948832458496000, 1115089078488795648000, 27290469545695931904000, 694002594415741341696000

Offset: 0

N. J. A. Sloane , Simon Plouffe, Robert G. Wilson v, Mira Bernstein

Eulerian permutations of the multiset {1,1,2,2,...,n+3,n+3} with n ascents.Eulerian permutations have the restriction that for all m, all integers between the two copies of m are less than m. In particular, the two 1s are always next to each other.

The sequence gives the Eulerian numbers <<3,0>>, <<4,1>>, <<5,2>>, <<6,3>>, ... (and in particular the offset is 0).

			For instance, a(1) = 22 because among the 7!! = 105 permutations of {1,1,2,2,3,3,4,4} selected according to the definition of Eulerian numbers of the second kind, only 22 contain n = 1 descent, namely : 11223443, 11224433, 11233244, 11233442, 11244233, 11332244, 11334422, 11442233, 12213344, 12233144, 12233441, 12244133, 13312244, 13344122, 14412233, 22113344, 22331144, 22334411, 22441133, 33112244, 33441122, 44112233. - _Jean-François Alcover_, Mar 28 2011

Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Math., 2nd edition; Addison-Wesley, 1994, pp. 270-271.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=0..99
L. Carlitz, Some numbers related to the Stirling numbers of the first and second kind, Univ. Beograd Publ. Elektrotehn. Fak. Ser. Mat. Fiz., Numbers 544-576 (1976): 49-55. [Annotated scanned copy. The triangle is A008517.]
I. Gessel and R. P. Stanley, Stirling polynomials, J. Combin. Theory, A 24 (1978), 24-33.
O. J. Munch, Om potensproduktsummer [Norwegian, English summary], Nordisk Matematisk Tidskrift, 7 (1959), 5-19. [Annotated scanned copy]
O. J. Munch, Om potensproduktsummer [ Norwegian, English summary ], Nordisk Matematisk Tidskrift, 7 (1959), 5-19. There are errors in the last two rows of his table.

3rd diagonal of A008517, third column of A112007.

b[1]=1; b[2]=22; b[n_] := b[n] = ((n-1)*(n-1)!*n^3 - (n+2)*(n+3)*b[n-2]*n + (n*(2*n+5)-4)*b[n-1]) / (n-1); a[n_] := b[n+1]; Table[a[n], {n, 0, 19}] (* Jean-François Alcover, Mar 23 2011, updated Oct 12 2015 *)

{a=vector(30,n,1);a[2]=22;for(n=3,#a,a[n]=(n-1)!*n^3+((n*(2*n+5)-4)*a[n-1] - n*(n+2)*(n+3)*a[n-2])/(n-1));a} \\ Uses offet 1 for technical reasons. - M. F. Hasler, Sep 19 2015

a(n)= (n+5)*a(n-1) + (n+1)*A002538(n+1), n>=1, a(0)=1.
Recurrence: (n-1)*n^2*a(n) = (n-1)*(3*n^3 + 12*n^2 + 6*n + 1)*a(n-1) - (n+1)*(3*n^4 + 15*n^3 + 13*n^2 - 15*n - 4)*a(n-2) + n*(n+1)^3*(n+2)*(n+3)*a(n-3). - Vaclav Kotesovec, May 24 2014
a(n) ~ n! * n^5 * log(n) * (log(n)*(1/2+181/(24*n)) + gamma*(1+181/(12*n)) - 2 - 65/(3*n)), where gamma is the Euler-Mascheroni constant (A001620). - Vaclav Kotesovec, May 24 2014

Formulas adapted for offset 0 by Vaclav Kotesovec, May 24 2014

More terms from M. F. Hasler, Sep 19 2015

A049218 Triangle T(n,k) of arctangent numbers: expansion of arctan(x)^n/n!.

Original entry on oeis.org

1, 0, 1, -2, 0, 1, 0, -8, 0, 1, 24, 0, -20, 0, 1, 0, 184, 0, -40, 0, 1, -720, 0, 784, 0, -70, 0, 1, 0, -8448, 0, 2464, 0, -112, 0, 1, 40320, 0, -52352, 0, 6384, 0, -168, 0, 1, 0, 648576, 0, -229760, 0, 14448, 0, -240, 0, 1, -3628800, 0, 5360256, 0, -804320, 0, 29568, 0, -330, 0, 1

Offset: 1

N. J. A. Sloane

|T(n,k)| gives the sum of the M_2 multinomial numbers (A036039) for those partitions of n with exactly k odd parts. E.g.: |T(6,2)| = 144 + 40 = 184 from the partitions of 6 with exactly two odd parts, namely (1,5) and (3,3), with M_2 numbers 144 and 40. Proof via the general Jabotinsky triangle formula for |T(n,k)| using partitions of n into k parts and their M_3 numbers (A036040). Then with the special e.g.f. of the (unsigned) k=1 column, f(x):= arctanh(x), only odd parts survive and the M_3 numbers are changed into the M_2 numbers. For the Knuth reference on Jabotinsky triangles see A039692. - Wolfdieter Lang, Feb 24 2005 [The first two sentences have been corrected thanks to the comment by José H. Nieto S. given below. - Wolfdieter Lang, Jan 16 2012]
|T(n,k)| gives the number of permutations of {1,2,...,n} (degree n permutations) with the number of odd cycles equal to k. E.g.: |T(5,3)|= 20 from the 20 degree 5 permutations with cycle structure (.)(.)(...). Proof: Use the cycle index polynomial for the symmetric group S_n (see the M_2 array A036039 or A102189) together with the partition interpretation of |T(n,k)| given above. - Wolfdieter Lang, Feb 24 2005 [See the following José H. Nieto S. correction. - Wolfdieter Lang, Jan 16 2012]
The first sentence of the above comment is inexact, it should be "|T(n,k)| gives the number of degree n permutations which decompose into exactly k odd cycles". The number of degree n permutations with k odd cycles (and, possibly, other cycles of even length) is given by A060524. - José H. Nieto S., Jan 15 2012
The unsigned triangle with e.g.f. exp(x*arctanh(z)) is the associated Jabotinsky type triangle for the Sheffer type triangle A060524. See the comments there. - Wolfdieter Lang, Feb 24 2005
Also the Bell transform of the sequence (-1)^(n/2)*A005359(n) without column 0. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 28 2016

			Triangle begins:
   1;
   0,   1;
  -2,   0,   1;
   0,  -8,   0,   1;
  24,   0, -20,   0,   1;
   0, 184,   0, -40,   0,   1;
  ...
O.g.f. for fifth subdiagonal: (24*t+16*t^2)/(1-t)^7 = 24*t + 184*t^2 + 784*t^3 + 2404*t^4 + ....

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 260.

Peter Bala, Diagonals of triangles with generating function exp(t*F(x)).
Steven Finch, Rounds, Color, Parity, Squares, arXiv:2111.14487 [math.CO], 2021.
Kruchinin Vladimir Victorovich, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.
Eric Weisstein's World of Mathematics, Mittag-Leffler Polynomial

Essentially same as A008309, which is the main entry for this sequence.

Row sums (unsigned) give A000246(n); signed row sums give A002019(n), n>=1. A137513.

A049218 := proc(n,k)(-1)^((3*n+k)/2) *add(2^(j-k)*n!/j! *stirling1(j,k) *binomial(n-1,j-1),j=k..n) ; end proc: # R. J. Mathar, Feb 14 2011
# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> `if`(n::odd, 0, (-1)^(n/2)*n!), 10); # Peter Luschny, Jan 28 2016

t[n_, k_] := (-1)^((3n+k)/2)*Sum[ 2^(j-k)*n!/j!*StirlingS1[j, k]*Binomial[n-1, j-1], {j, k, n}]; Flatten[ Table[ t[n, k], {n, 1, 11}, {k, 1, n}]] (* Jean-François Alcover, Dec 06 2011, after Vladimir Kruchinin *)
BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len-1}, {k, 0, len-1}]];
rows = 12;
M = BellMatrix[If[OddQ[#], 0, (-1)^(#/2)*#!]&, rows];
Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 23 2018, after Peter Luschny *)

T(n,k)=polcoeff(serlaplace(atan(x)^k/k!), n)

E.g.f.: arctan(x)^k/k! = Sum_{n>=0} T(n, k) x^n/n!.
T(n,k) = ((-1)^((3*n+k)/2)*n!/2^k)*Sum_{i=k..n} 2^i*binomial(n-1,i-1)*Stirling1(i,k)/i!. - Vladimir Kruchinin, Feb 11 2011
E.g.f.: exp(t*arctan(x)) = 1 + t*x + t^2*x^2/2! + t*(t^2-2)*x^3/3! + .... The unsigned row polynomials are the Mittag-Leffler polynomials M(n,t/2). See A137513. The compositional inverse (with respect to x) (x-t/2*log((1+x)/(1-x)))^(-1) = x/(1-t) + 2*t/(1-t)^4*x^3/3!+ (24*t+16*t^2)/(1-t)^7*x^5/5! + .... The rational functions in t generate the (unsigned) diagonals of the table. See the Bala link. - Peter Bala, Dec 04 2011

Additional comments from Michael Somos

A053567 Stirling numbers of first kind, s(n+5, n).

Original entry on oeis.org

-120, 1764, -13132, 67284, -269325, 902055, -2637558, 6926634, -16669653, 37312275, -78558480, 156952432, -299650806, 549789282, -973941900, 1672280820, -2792167686, 4546047198, -7234669596, 11276842500, -17247104875, 25922927745, -38343278610, 55880640270

Offset: 1

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Jan 17 2000

a(n) is equal to (-1)^n times the sum of the products of each distinct grouping of 5 members of the set {1, 2, 3, ..., n + 4}. So, a(1) = (-1)*1*2*3*4*5 = -120, and a(2) = 1*2*3*4*5 + 1*2*3*4*6 + 1*2*3*5*6 + 1*2*4*5*6 + 1*3*4*5*6 + 2*3*4*5*6 = 1764. See comment at A001303. - Greg Dresden, Aug 26 2019

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 833.

Vincenzo Librandi, Table of n, a(n) for n = 1..200
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].
G. C. Greubel, A Note on Jain basis functions, arXiv:1612.09385 [math.CA], 2016.
Index entries for linear recurrences with constant coefficients, signature (-11,-55,-165,-330,-462,-462,-330,-165,-55,-11,-1).

Next |Stirling1| diagonal A112002, 5th diagonal of A130534.

[(-1)^n*Binomial(n+5, 6)*Binomial(n+5, 2)*(3*n^2+23*n+38)/8: n in [1..30]]; // Vincenzo Librandi, Jun 09 2011

A053567 := proc(n) (-1)^(n+1)*combinat[stirling1](n+5,n) ; end proc: # R. J. Mathar, Jun 08 2011

Table[StirlingS1[n+5,n](-1)^(n-1),{n,30}] (* Harvey P. Dale, Sep 21 2011 *)
(* or *)
CoefficientList[Series[-x*(120 - 444*x + 328*x^2 - 52*x^3 + x^4)/(1+x)^11, {x, 0, 27}], x] (* Georg Fischer, May 19 2019 *)

a(n) = (-1)^(n-1)*stirling(n+5, n, 1); \\ Michel Marcus, Aug 29 2017

[stirling_number1(n,n-5)*(-1)^(n+1) for n in range(6, 26)] # Zerinvary Lajos, May 16 2009

a(n) = (-1)^n*binomial(n+5, 6)*binomial(n+5, 2)*(3*n^2 + 23*n + 38)/8.
G.f.: -x*(120 - 444*x + 328*x^2 - 52*x^3 + x^4)/(1+x)^11. See row k=4 of triangle A112007 for the coefficients. [G.f. corrected by Georg Fischer, May 19 2019]
E.g.f. with offset 5: exp(x)*(Sum_{m=0..5} A112486(5, m)*(x^(5+m)/(5+m)!).
a(n) = (f(n+4, 5)/10!)*Sum_{m=0..min(5, n-1)} A112486(5, m)*f(10, 5-m)*f(n-1, m)), with the falling factorials f(n, m):=n*(n-1)*, ..., *(n-(m-1)). From the e.g.f.

Definition edited by Eric M. Schmidt, Aug 29 2017

Incorrect formula removed by Greg Dresden, Aug 26 2019

A288876 a(n) = binomial(n+4, n)^2. Square of the fifth diagonal sequence of A007318 (Pascal). Fifth diagonal sequence of A008459.

Original entry on oeis.org

1, 25, 225, 1225, 4900, 15876, 44100, 108900, 245025, 511225, 1002001, 1863225, 3312400, 5664400, 9363600, 15023376, 23474025, 35820225, 53509225, 78411025, 112911876, 160022500, 223502500, 308002500, 419225625, 564110001, 751034025, 990046225, 1293121600, 1674446400, 2150733376

Offset: 0

Wolfdieter Lang, Jul 27 2017

This is also the square of the fifth (k = 4) column sequence (without leading zeros) of the Pascal triangle A007318. For the triangle with the squares of the entries of Pascal's triangle see A008459.
For the square of the (d+1)-th diagonal sequence of A007318, PD2(d,n) = binomial(d + n, n)^2, d >= 0, one finds the o.g.f. GPD2(d, x) = Sum_{n>=0} PD2(d,n)*x^n in the following way. Compute the compositional inverse (Lagrange inversion formula) of y(t,x) = x*(1 - t/(1-x)) w.r.t. x, that is x = x(t,y). Then -log(1 - x(t,y)) = Sum_{d=0} y^(d+1)/(d+1)*GPD2(d, x). The r.h.s. can be called the logarithmic generating function (l.g.f.) of the o.g.f.s of the square of the diagonals of Pascal's triangle.
This computation was inspired by an article by P. Bala (see a link in A112007) on the diagonal sequences of special Sheffer triangles (1, f(t)) (Sheffer triangles are there called exponential Riordan triangles, and f is called F). This can be generalized to Sheffer (g, f). For general Riordan triangles R = (G(x), F(x)) a similar analysis can be done. The present entry is then obtained for example of the Pascal triangle P = (1/(1-x), x/(1-x)).
The o.g.f.s for the square of the diagonals of Pascal's triangle turn out to be GPD2(d, x) = P(d,x)/(1 - x)^(2*d+1), with the numerator polynomials given by row n of triangle A008459 (squares of the entries of Pascal's triangle): P(d, x) = Sum_{k=0..d} A008459(d, k)*x^k.

Cf. A007318, A008459.

The squares of the first diagonals are in A000012, A000290(n+1), A000537, A001249 (for d = 0..3).

[Binomial(n+4, n)^2: n in [0..30]]; // Vincenzo Librandi, Aug 02 2017

Table[Binomial[n + 4, n]^2, {n, 0, 30}] (* Michael De Vlieger, Jul 30 2017 *)

a(n) = binomial(n+4, n)^2 \\ Felix Fröhlich, Jul 31 2017

a(n) = binomial(n+4, n)^2, n >= 0.
O.g.f.: (1 + 16*x + 36*x^2 + 16*x^3 + x^4)/(1 - x)^9. (See a comment above and row n=4 of A008459.)
E.g.f: exp(x)*(1 + 24*x + 176*x^2/2! + 624*x^3/3! + 1251*x^4/4!+ 1500*x^5/5!+ 1070*x^6/6! + 420*x^7/7! + 70*x^8/8!), computed from the o.g.f with the formulas (23) - (25) of the W. Lang link given in A060187.
From Amiram Eldar, Sep 20 2022: (Start)
Sum_{n>=0} 1/a(n) = 160*Pi^2/3 - 1576/3.
Sum_{n>=0} (-1)^n/a(n) = 512*log(2)/3 - 352/3. (End)

A075856 Triangle formed from coefficients of the polynomials p(1)=x, p(n+1) = (n + x(n+1))p(n) + xx(d/dx)p(n).

Original entry on oeis.org

1, 1, 3, 2, 10, 15, 6, 40, 105, 105, 24, 196, 700, 1260, 945, 120, 1148, 5068, 12600, 17325, 10395, 720, 7848, 40740, 126280, 242550, 270270, 135135, 5040, 61416, 363660, 1332100, 3213210, 5045040, 4729725, 2027025

Offset: 1

F. Chapoton, Oct 15 2002

Constant terms of polynomials related to Ramanujan psi polynomials (see Zeng reference).

			Triangle begins
    1;
    1,    3;
    2,   10,   15;
    6,   40,  105,   105;
   24,  196,  700,  1260,   945;
  120, 1148, 5068, 12600, 17325, 10395;
  ...
p(1) = x, p(2) = 3*x^2 + x, p(3) = 15*x^3 + 10*x^2 + 2*x, etc. - _Michael Somos_, Mar 17 2011

P. Bala, Diagonals of triangles with generating function exp(t*F(x)).
J. Fernando Barbero G., Jesús Salas, and Eduardo J. S. Villaseñor, Bivariate Generating Functions for a Class of Linear Recurrences. I. General Structure, arXiv:1307.2010 [math.CO], 2013.
Brian Drake, Ira M. Gessel, and Guoce Xin, Three Proofs and a Generalization of the Goulden-Litsyn-Shevelev Conjecture on a Sequence Arising in Algebraic Geometry, J. of Integer Sequences, Vol. 10 (2007), #07.3.7.
Dominique Dumont and Armand Ramamonjisoa, Grammaire de Ramanujan et Arbres de Cayley, Electr. J. Combinatorics, Volume 3, Issue 2 (1996) R17 (see page 16).
H. W. Gould, A Set of Polynomials Associated with the Higher Derivatives of y = x^x, Rocky Mountain J. Math. Volume 26, Number 2 (1996), 615-625.
M. Josuat-Vergès, Derivatives of the tree function, arXiv preprint arXiv:1310.7531 [math.CO], 2013.
S. Ramanujan, Notebook entry
P. W. Shor, Problem 78-6: A combinatorial identity, in Problems and Solutions column, SIAM Review; problem in 20, p. 394 (1978); solution in 21, pp. 258-260 (1979). [N. Sato, Feb 19 2010]
P. W. Shor, A = B (but not quite); 3-d array with multiple recurrences, MathOverflow, Nov 2010-Nov 2011.
J. Zeng, A Ramanujan sequence that refines the Cayley formula for trees, manuscript, 1996.
J. Zeng, A Ramanujan sequence that refines the Cayley formula for trees, Ramanujan J., 3(1999) 1, 45-54.

Cf. A000142, A000312, A001147, A054589. A185164, A209937.

See A239098 for another version.

p[1] = x; p[n_] := p[n] = (n - 1 + x*n)*p[n - 1] + x*x*D[p[n - 1], x]; Flatten[Rest[CoefficientList[#1, x]] & /@ Table[p[n], {n, 8}]] (* Jean-François Alcover, May 31 2011 *)

{T(n, k) = if( k<1 || nMichael Somos, Mar 17 2011 */

T(n, k) = (n-1) * T(n-1, k) + (n+k-1) * T(n-1, k-1). - Michael Somos, Mar 17 2011
G.f.: A(x, t) = Sum_{n>0} p[n] t^n / n! satisfies (dA / dt) * (x + t - 1) = x * (1 + A)^2 * (x * (1 + A) - 1). - Michael Somos, Mar 17 2011
T(n, 1) = (n-1)! = A000142(n-1). T(n, n) = A001147(n). Sum_{k>0} T(n, k) = n^n = A000312(n). Sum_{k>0} T(n, k) x^k = p[n].
From Peter Bala, Mar 14 2012: (Start)
This triangle is A185164 read by diagonals.
Let F(x) = x + (1-x)*log(1-x). The e.g.f. is given by the compositional inverse
(x - t*F(x))^(-1) = x + t*x^2/2! + (t + 3*t^2)x^3/3! + (2*t + 10*t^2 + 15*t^3)*x^4/4! + ....
Let f(x) = 1/log(1+x) and define inductively D^(n+1)(f(x)) = f(x)*(d/dx)(D^n(f(x))) with D^(0)f(x) = f(x). Then D^(n)f = (-1)^n*Sum_{k = 1..n} T(n,k)*f^(n-k)/((1+x)^n*f^(2n+1)).
(End)

A137513 Triangle read by rows: the coefficients of the Mittag-Leffler polynomials.

Original entry on oeis.org

1, 0, 2, 0, 0, 4, 0, 4, 0, 8, 0, 0, 32, 0, 16, 0, 48, 0, 160, 0, 32, 0, 0, 736, 0, 640, 0, 64, 0, 1440, 0, 6272, 0, 2240, 0, 128, 0, 0, 33792, 0, 39424, 0, 7168, 0, 256, 0, 80640, 0, 418816, 0, 204288, 0, 21504, 0, 512, 0, 0, 2594304, 0, 3676160, 0, 924672, 0, 61440, 0, 1024

Offset: 1

Roger L. Bagula, Apr 23 2008

Previous name was: Triangle read by rows: coefficients of the expansion of a polynomial related to the Poisson kernel: p(t,r) = ((1 + t)/(1 - t))^x: r*Exp(i*theta) -> t.
General relation is that Poisson's kernel is the real part of this type of function (page 31 Hoffman reference).
The row polynomials of this table are the Mittag-Leffler polynomials M(n,t), a polynomial sequence of binomial type [Roman, Chapter 4, Section 1.6]. The first few values are M(0,t) = 1, M(1,t) = 2*t, M(2,t) = 4*t^2, M(3,t) = 4*t+8*t^3. The polynomials M(n,t/2) are the (unsigned) row polynomials of A049218. - Peter Bala, Dec 04 2011
Also the Bell transform of the sequence "a(n) = 2*n! if n is even else 0". For the definition of the Bell transform see A264428. - Peter Luschny, Jan 28 2016

			{1},
{0, 2},
{0, 0, 4},
{0, 4, 0, 8},
{0, 0, 32, 0, 16},
{0, 48, 0, 160, 0, 32},
{0, 0, 736, 0, 640, 0, 64},
{0, 1440, 0, 6272, 0, 2240, 0, 128},
{0, 0, 33792, 0, 39424, 0, 7168, 0, 256},
{0, 80640, 0, 418816, 0, 204288, 0, 21504, 0, 512},
{0, 0, 2594304, 0, 3676160,0, 924672, 0, 61440, 0, 1024}

Kenneth Hoffman, Banach Spaces of Analytic Functions, Dover, New York, 1962, page 30.
Thomas McCullough, Keith Phillips, Foundations of Analysis in the Complex Plane, Holt, Reinhart and Winston, New York, 1973, 215.
S. Roman, The Umbral Calculus: Dover Publications, New York (2005).

Peter Bala, Diagonals of triangles with generating function exp(t*F(x)).
H. Bateman, The Polynomial of Mittag-Leffler. PNAS, 26 (8), 1940, 491-496.
Eric Weisstein's World of Mathematics, Mittag-Leffler Polynomial

Cf. A049218, A098558 (row sums).

A137513_row := proc(n) `if`(n=0,1,2*x*hypergeom([1-n,1-x],[2],2));
PolynomialTools[CoefficientList](expand(n!*simplify(%,hypergeom)),x) end:
seq(A137513_row(n),n=0..10): ListTools[FlattenOnce]([%]); # Peter Luschny, Jan 28 2016
# Alternatively, using the function BellMatrix defined in A264428:
BellMatrix(n -> `if`(n::odd, 0, 2*n!), 9); # Peter Luschny, Jan 28 2016

p[t_] = ((1 + t)/(1 - t))^x; Table[ ExpandAll[n! * SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]], {n, 0, 10}]; a = Table[ CoefficientList[n!*SeriesCoefficient[ FullSimplify[Series[p[t], {t, 0, 30} ]], n], x], {n, 0, 10}]; Flatten[a]
MLP[n_] := Sum[Binomial[n, k]*2^k*FactorialPower[n - 1, n - k]* FactorialPower[x, k] // FunctionExpand, {k, 0, n}]; Table[ CoefficientList[MLP[n], x], {n, 0, 9}] // Flatten (* or: *)
MLP[0] = 1; MLP[n_] := 2x*n!*Hypergeometric2F1[1-n, 1-x, 2, 2]; Table[ CoefficientList[MLP[n], x], {n, 0, 9}] // Flatten (* or: *)
BellMatrix[If[OddQ[#], 0, 2*#!]&, 9] (* in triangular matrix form, using Peter Luschny's BellMatrix function defined in A264428 *) (* Jean-François Alcover, Jan 29 2016 *)

MLP = lambda n: sum(binomial(n, k)*2^k*falling_factorial(n-1, n-k)* falling_factorial(x, k) for k in (0..n)).expand()
def A137513_row(n): return MLP(n).list()
for n in (0..9): A137513_row(n) # Peter Luschny, Jan 28 2016

From Peter Bala, Dec 04 2011: (Start)
T(n,k) = (-1)^k*(n-1)!*Sum_{i=k..n} (-2)^i*binomial(n,i)/(i-1)!*|Stirling1(i,k)|.
E.g.f.: Sum_{n>=0} M(n,t)*x^n/n! = exp(t*log((1+x)/(1-x))) = ((1+x)/(1-x))^t = exp(2*t*atanh(x)) = 1 + (2*t)*x + (4*t^2)*x^2/2! + (4*t+8*t^3)*x^3/3! + ....
M(n,t) = (n-1)!*Sum_{k = 1..n} k*2^k*binomial(n,k)*binomial(t,k), for n>=1.
Recurrence relation: M(n+1,t) = 2*t*Sum_{k = 0..floor(n/2)} (n!/(n-2*k)!)* M(n-2*k,t), with M(0,t) = 1.
The o.g.f. for the n-th diagonal of the table is a rational function in t, given by the coefficient of x^n/n! in the expansion (with respect to x) of the compositional inverse (x-t*log((1+x)/(1-x)))^(-1) = x/(1-2*t) + 4*t/(1-2*t)^4*x^3/3! + (48*t+64*t^2)/(1-2*t)^7*x^5/5! + ...; for example, the o.g.f. for the fifth subdiagonal is (48*t+64*t^2)/(1-2*t)^7 = 48*t + 736*t^2 + 6272*t^3+ .... See the Bala link.
(End)
The row polynomials satisfy M(n, t+1) - M(n, t-1) = 2*n*M(n, t)/t. - Peter Bala, Nov 16 2016

Edited and new name by Peter Luschny, Jan 28 2016

A185164 Coefficients of a set of polynomials associated with the derivatives of x^x.

Original entry on oeis.org

1, 1, 2, 3, 6, 10, 24, 40, 15, 120, 196, 105, 720, 1148, 700, 105, 5040, 7848, 5068, 1260, 40320, 61416, 40740, 12600, 945, 362880, 541728, 363660, 126280, 17325, 3628800, 5319072, 3584856, 1332100, 242550, 10395, 39916800, 57545280, 38764440, 15020720, 3213210, 270270

Offset: 2

Peter Bala, Mar 12 2012

Gould shows that the derivatives of x^x are given by (d/dx)^n(x^x) = (x^x)*Sum_{k = 0..n} (-1)^k*binomial(n,k)*(1 + log(x))^(n-k)*x^(-k)*R(k,x), where R(n,x) is a polynomial in x of degree floor(n/2). The first few values are R(0,x) = 1, R(1,x) = 0, R(2,x) = x, R(3,x) = x and R(4,x) = 2*x + 3*x^2. The coefficients of these polynomials are listed in the table for n >= 2. Gould gives an explicit formula for R(n,x) as a triple sum, and also an expression in terms of the Comtet numbers A008296.

This table read by diagonals gives A075856.

			Triangle begins
n\k.|.....1.....2.....3.....4
= = = = = = = = = = = = = = =
..2.|.....1
..3.|.....1
..4.|.....2.....3
..5.|.....6....10
..6.|....24....40....15
..7.|...120...196...105
..8.|...720..1148...700...105
..9.|..5040..7848..5068..1260
...
Fourth derivative of x^x:
x^(-x)*(d/dx)^4(x^x) = (1+log(x))^4 + C(4,2)/x^2*(1+log(x))^2*x - C(4,3)/x^3*(1+log(x)) + C(4,4)/x^4*(2*x + 3*x^2).
Example of recurrence relation for table entries:
T(7,2) = 4*T(6,2) + 6*T(5,1) = 4*40 + 6*6 = 196.

Robert Israel, Table of n, a(n) for n = 2..10001 (rows 2 to 200, flattened)
Peter Bala, Diagonals of triangles with generating function exp(t*F(x)).
H. W. Gould, A Set of Polynomials Associated with the Higher Derivatives of y = x^x, Rocky Mountain J. Math. Volume 26, Number 2 (1996), 615-625.

Cf. A008296, A075856, A203852 (row sums).

T[2,1]:= 1:
for n from 3 to 15 do
  for k from 1 to floor(n/2) do
    T[n,k]:= (n-1-k)*`if`(k<= floor((n-1)/2),T[n-1,k],0) + `if`(k>=2 and k-1 <= floor((n-2)/2),(n-1)*T[n-2,k-1],0)
od od:
seq(seq(T[n,k],k=1..floor(n/2)),n=2..15); # Robert Israel, Jan 13 2016

m = 14; F = Exp[t (x + (1-x) Log[1-x])];
cc = CoefficientList[# + O[t]^m, t]& /@ CoefficientList[F + O[x]^m, x]* Range[0, m - 1]!;
Rest /@ Drop[cc, 2] (* Jean-François Alcover, Jun 26 2019 *)

# uses[bell_transform from A264428]
# Computes the full triangle for n>=0 and 0<=k<=n.
def A185164_row(n):
    g = lambda k: factorial(k-1) if k>0 else 0
    s = [g(k) for k in (0..n)]
    return bell_transform(n, s)
[A185164_row(n) for n in (0..10)] # Peter Luschny, Jan 13 2016

Recurrence relation: T(n+1,k) = (n - k)*T(n,k) + n*T(n-1,k-1).
The diagonal entries D(n,k) := T(n+k,k) satisfy the recurrence D(n+1,k) = n*D(n,k) + (n + k)*D(n,k-1) so this table read by diagonals is A075856.
E.g.f.: F(x,t) = exp(t*(x + (1 - x)*log(1 - x))) = Sum_{n = 0..oo} R(n,t)*x^n/n! = 1 + t*x^2/2! + t*x^3/3! + (2*t + 3*t^2)*x^4/4! + .... The e.g.f. F(x,t) satisfies the partial differential equation (1 - x)*dF/dx + t*dF/dt = x*t*F.
This gives the recurrence relation for the row generating polynomials: R(n+1,x) = n*R(n,x) - x*d/dx(R(n,x)) + n*x*R(n-1,x) for n >= 1, with initial conditions R(0,x) = 1, R(1,x) = 0.
The e.g.f. for the triangle read by diagonals is given by the series reversion (with respect to x) (x - t*(x + (1 - x)*log(1 - x)))^(-1) = x + t*x^2/2! + (t + 3*t^2)x^3/3! + (2*t + 10*t^2 + 15*t^3)*x^4/4! + ....
Diagonal sums: Sum_{k = 1..n} T(n+k,k) = n^n , n >= 1.
Row sums A203852.
Also the Bell transform of the sequence g(k) = (k-1)! if k>0 else 0. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 13 2016

More terms from Jean-François Alcover, Jun 26 2019

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A061356 Triangle read by rows: T(n, k) is the number of labeled trees on n nodes with maximal node degree k (0 < k < n).

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A111999 T(n, k) = [x^k] (-1)^n*Sum_{k=0..n} E2(n, n-k)*(1+x)^(n-k) where E2(n, k) are the second-order Eulerian numbers. Triangle read by rows, T(n, k) for n >= 1 and 0 <= k <= n.

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A112486 Coefficient triangle for polynomials used for e.g.f.s for unsigned Stirling1 diagonals.

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A002539 Eulerian numbers of the second kind: <>.

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A049218 Triangle T(n,k) of arctangent numbers: expansion of arctan(x)^n/n!.

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A053567 Stirling numbers of first kind, s(n+5, n).

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A111999 T(n, k) = [x^k] (-1)^nSum_{k=0..n} E2(n, n-k)(1+x)^(n-k) where E2(n, k) are the second-order Eulerian numbers. Triangle read by rows, T(n, k) for n >= 1 and 0 <= k <= n.

A075856 Triangle formed from coefficients of the polynomials p(1)=x, p(n+1) = (n + x(n+1))p(n) + xx(d/dx)p(n).