A008276 -id:A008276 - cp's OEIS frontend

A008278 Reflected triangle of Stirling numbers of 2nd kind, S(n,n-k+1), n >= 1, 1 <= k <= n.

1, 1, 1, 1, 3, 1, 1, 6, 7, 1, 1, 10, 25, 15, 1, 1, 15, 65, 90, 31, 1, 1, 21, 140, 350, 301, 63, 1, 1, 28, 266, 1050, 1701, 966, 127, 1, 1, 36, 462, 2646, 6951, 7770, 3025, 255, 1, 1, 45, 750, 5880, 22827, 42525, 34105, 9330, 511, 1

Offset: 1

N. J. A. Sloane

The n-th row also gives the coefficients of the sigma polynomial of the empty graph \bar K_n. - Eric W. Weisstein, Apr 07 2017
The n-th row also gives the coefficients of the independence polynomial of the (n-1)-triangular honeycomb bishop graph. - Eric W. Weisstein, Apr 03 2018
From Gus Wiseman, Aug 11 2020: (Start)
Conjecture: also the number of divisors of the superprimorial A006939(n - 1) that have 0 <= k <= n distinct prime factors, all appearing with distinct multiplicities. For example, row n = 4 counts the following divisors of 360:
  1  2  12  360
     3  18
     4  20
     5  24
     8  40
     9  45
        72
Equivalently, T(n,k) is the number of length-n vectors 0 <= v_i <= i with k nonzero values, all of which are distinct.
Crossrefs:
A006939 lists superprimorials or Chernoff numbers.
A022915 counts permutations of prime indices of superprimorials.
A076954 can be used instead of A006939.
A130091 lists numbers with distinct prime multiplicities.
A181796 counts divisors with distinct prime multiplicities.
A336420 is the version counting all prime factors, not just distinct ones.
Cf. A000005, A027423, A095149, A124010, A317829, A327498, A336419, A336421.
(End)
From Leonidas Liponis, Aug 26 2024: (Start)
It appears that this sequence is related to the combinatorial form of Faà di Bruno's formula. Specifically, the number of terms for the n-th derivative of a composite function y = f(g(x)) matches the number of partitions of n.
For example, consider the case where g(x) = e^x, in which all derivatives of g(x) are equal. The first 5 rows of A008278 appear as the factors of derivatives of f(x), highlighted here in brackets:
          dy/dx = [ 1 ] * f'(e^x) * e^x
          d^2y/dx^2 = [ 1 ] * f''(e^x) * e^{2x} + [ 1 ] * f'(e^x) * e^x
          d^3y/dx^3 = [ 1 ] * f'''(e^x) * e^{3x} + [ 3 ] * f''(e^x) * e^{2x} + [ 1 ] * f'(e^x) * e^x
          d^4y/dx^4 = [ 1 ] * f''''(e^x) * e^{4x} + [ 6 ] * f'''(e^x) * e^{3x} + [ 7 ] * f''(e^x) * e^{2x} + [ 1 ] * f'(e^x) * e^x
          d^5y/dx^5 = [ 1 ] * f'''''(e^x) * e^{5x} + [ 10 ] * f''''(e^x) * e^{4x} + [ 25 ] * f'''(e^x) * e^{3x} + [ 15 ] * f''(e^x) * e^{2x} + [ 1 ] * f'(e^x) * e^x
This pattern is observed in Mathematica for the first 10 cases, using the code below.
(End)

			The e.g.f. of [0,0,1,7,25,65,...], the k=3 column of A008278, but with offset n=0, is exp(x)*(1*(x^2)/2! + 4*(x^3)/3! + 3*(x^4)/4!).
Triangle starts:
  1;
  1,  1;
  1,  3,   1;
  1,  6,   7,    1;
  1, 10,  25,   15,    1;
  1, 15,  65,   90,   31,    1;
  1, 21, 140,  350,  301,   63,    1;
  1, 28, 266, 1050, 1701,  966,  127,   1;
  1, 36, 462, 2646, 6951, 7770, 3025, 255, 1;
  ...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 835.
F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 223.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, 2nd ed., 1994.

T. D. Noe, Rows n = 0..100 of triangle, 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].
Noureddine Chair, Exact two-point resistance, and the simple random walk on the complete graph minus N edges, Ann. Phys. 327, No. 12, 3116-3129 (2012), eq. (27).
Xi Chen, Bishal Deb, Alexander Dyachenko, Tomack Gilmore, and Alan D. Sokal, Coefficientwise total positivity of some matrices defined by linear recurrences, arXiv:2012.03629 [math.CO], 2020.
T. Copeland, Generators, Inversion, and Matrix, Binomial, and Integral Transforms, 2015
U. N. Katugampola, A new Fractional Derivative and its Mellin Transform, arXiv preprint arXiv:1106.0965 [math.CA], 2011.
Eric Weisstein's World of Mathematics, Bell Polynomial
Eric Weisstein's World of Mathematics, Empty Graph
Eric Weisstein's World of Mathematics, Independence Polynomial
Eric Weisstein's World of Mathematics, Sigma Polynomial
Eric Weisstein's World of Mathematics, Stirling Number of the Second Kind

See A008277 and A048993, which are the main entries for this triangle of numbers.

Cf. A094262, A008277, A008276, A003422, A000166, A000110, A000204, A000045, A000108, A213735.

a008278 n k = a008278_tabl !! (n-1) !! (k-1)
a008278_row n = a008278_tabl !! (n-1)
a008278_tabl = iterate st2 [1] where
  st2 row = zipWith (+) ([0] ++ row') (row ++ [0])
            where row' = reverse $ zipWith (*) [1..] $ reverse row
-- Reinhard Zumkeller, Jun 22 2013

rows = 10; Flatten[Table[StirlingS2[n, k], {n, 1, rows}, {k, n, 1, -1}]] (* Jean-François Alcover, Nov 17 2011, *)
Table[CoefficientList[x^n BellB[n, 1/x], x], {n, 10}] // Flatten (* Eric W. Weisstein, Apr 05 2017 *)
n = 5; Grid[Prepend[Transpose[{Range[1, n], Table[D[f[Exp[x]], {x, i}], {i, 1, n}]}], {"Order","Derivative"}], Frame -> All, Spacings -> {2, 1}] (* Leonidas Liponis, Aug 27 2024 *)

for(n=1,10,for(k=1,n,print1(stirling(n,n-k+1,2),", "))) \\ Hugo Pfoertner, Aug 30 2020

T(n, k)=0 if n < k, T(n, 0)=0, T(1, 1)=1, T(n, k) = (n-k+1)*T(n-1, k-1) + T(n-1, k) otherwise.
O.g.f. for the k-th column: 1/(1-x) if k=1 and A(k,x):=((x^k)/(1-x)^(2*k+1))*Sum_{m=0..k-1} A008517(k,m+1)*x^m if k >= 2. A008517 is the second-order Eulerian triangle. Cf. p. 257, eq. (6.43) of the R. L. Graham et al. book. - Wolfdieter Lang, Oct 14 2005
E.g.f. for the k-th column (with offset n=0): E(k,x):=exp(x)*Sum_{m=0..k-1} A112493(k-1,m)*(x^(k-1+m))/(k-1+m)! if k >= 1. - Wolfdieter Lang, Oct 14 2005
a(n) = abs(A213735(n-1)). - Hugo Pfoertner, Sep 07 2020

Name edited by Gus Wiseman, Aug 11 2020

A082161 Number of deterministic completely defined initially connected acyclic automata with 2 inputs and n transient unlabeled states (and a unique absorbing state).

Original entry on oeis.org

1, 3, 16, 127, 1363, 18628, 311250, 6173791, 142190703, 3737431895, 110577492346, 3641313700916, 132214630355700, 5251687490704524, 226664506308709858, 10568175957745041423, 529589006347242691143, 28395998790096299447521

Offset: 1

Valery A. Liskovets, Apr 09 2003

Coefficients T_2(n,k) form the array A082169. These automata have no nontrivial automorphisms (by states).

Also counts the relaxed compacted binary trees of size n. A relaxed compacted binary tree of size n is a directed acyclic graph consisting of a binary tree with n internal nodes, one leaf, and n pointers. It is constructed from a binary tree of size n, where the first leaf in a post-order traversal is kept and all other leaves are replaced by pointers. These links may point to any node that has already been visited by the post-order traversal. See the Genitrini et al. link. - Michael Wallner, Apr 20 2017

			a(2)=3 since the following transition diagrams represent all three initially connected acyclic automata with two input letters x and y, two transient states 1 (initial) and 2 and the absorbing state 0:
  1 == x, y==> 2 == x, y ==> 0 == x, y ==> 0, 1 -- x --> 2 == x, y ==> 0 == x, y ==> 0
  1 -- y --> 0
and the last one with x and y interchanged.

Roland Bacher and Christophe Reutenauer, The number of right ideals of given codimension over a finite field, in Noncommutative Birational Geometry, Representations and Combinatorics, edited by Arkady. Berenstein and Vladimir. Retakha, Contemporary Mathematics, Vol. 592, 2013.

Vaclav Kotesovec, Table of n, a(n) for n = 1..350
David Callan, A determinant of Stirling cycle numbers counts unlabeled acyclic single-source automata, arXiv:0704.0004 [math.CO], 2007.
Manosij Ghosh Dastidar and Michael Wallner, Asymptotics of relaxed k-ary trees, arXiv:2404.08415 [math.CO], 2024. See p. 1.4.
Andrew Elvey Price, Wenjie Fang, and Michael Wallner, Compacted binary trees admit a stretched exponential, arXiv:1908.11181 [math.CO], 2019-2020; J. Combin. Theory Ser. A 177 (2021), Paper No. 105306, 40 pp.
Antoine Genitrini, Bernhard Gittenberger, Manuel Kauers and Michael Wallner, Asymptotic enumeration of compacted binary trees of bounded right height, arXiv:1703.10031 [math.CO], 2017; J. Combin. Theory Ser. A 172 (2020), 105177, 49 pp.
Valery A. Liskovets, Exact enumeration of acyclic automata, Proc. 15th Conf. "Formal Power Series and Algebr. Combin. (FPSAC'03)", 2003.
Valery A. Liskovets, Exact enumeration of acyclic deterministic automata, Discrete Appl. Math., 154, No.3 (2006), 537-551.
Fedor Petrov, On a generating function and vector ν of length n, answer to question on MathOverflow (2024).
Michael Wallner, A bijection of plane increasing trees with relaxed binary trees of right height at most one, arXiv:1706.07163 [math.CO], 2017-2018; Theoret. Comput. Sci. 755 (2019), 1-12.

Cf. A008276, A082157, A082169, A102086, A102316, A254789.

a[n_]:= a[n]= If[n==0, 1, Coefficient[1-Sum[a[k]*x^k*Product[1-j*x, {j, 1, k+1}], {k, 0, n-1}], x, n]];
Table[a[n], {n, 18}] (* Jean-François Alcover, Dec 15 2014, after Paul D. Hanna *)

{a(n)=if(n==0,1,polcoeff(1-sum(k=0,n-1,a(k)*x^k*prod(j=1,k+1,1-j*x+x*O(x^n))),n))} \\ Paul D. Hanna, Jan 07 2005

{a(n)=local(A);if(n<1,0,A=x+x*O(x^n); for(k=0,n,A+=polcoeff(A,k)*x^k*(1-prod(i=1,k+1,1-i*x))); polcoeff(A,n))} /* Michael Somos, Jan 16 2005 */

upto(n) = my(v=vector(n+1, i, i==1)); for(i=1, n, for(j=i+1, n+1, v[j] += i*v[j-1])); v[2..#v] \\ Mikhail Kurkov, Oct 25 2024

from functools import cache
@cache
def b(n, k):
    if n == 0: return k + 1
    return sum(b(j, k)*b(n-j-1, k+j) for j in range(n))
def A082161(n): return b(n, 0)
print([A082161(n) for n in range(1, 19)]) # G. C. Greubel, Jan 18 2024

a(n) = c_2(n)/(n-1)! where c_2(n) = T_2(n, 1) - Sum_{j=1..n-1} binomial(n-1, j-1)*T_2(n-j, j+1)*c_2(j), and T_2(0, k) = 1, T_2(n, k) = Sum_{i=0..n-1} binomial(n, i)*(-1)^(n-i-1)*(i+k)^(2*n-2*i)*T_2(i, k), n > 0.
Equals column 0 of triangle A102086. Also equals main diagonal of A102316: a(n) = A102086(n, 0) = A102316(n, n). - Paul D. Hanna, Jan 07 2005
G.f.: 1 = Sum_{n>=0} a(n)*x^n*prod_{k=1, n+1} (1-k*x) for n>0 with a(0)=1. a(n) = -Sum_{k=1, [(n+1)/2]} A008276(n-k+1, k)*a(n-k) where A008276 is Stirling numbers of the first kind. Thus G.f.: 1 = (1-x) + 1*x*(1-x)(1-2x) + 3*x^2*(1-x)(1-2x)(1-3x) + ... + a(n)*x^n*(1-x)(1-2x)(1-3x)*..*(1-(n+1)*x) + ... with a(0)=1. - Paul D. Hanna, Jan 14 2005
a(n) is the determinant of the n X n matrix with (i, j) entry = StirlingCycle[i+1, 2i-j]. - David Callan, Jul 20 2005
a(n) = b(n,0) where b(0,p) = p+1 and b(n+1,p) = Sum_{i=0..n} b(i,p)*b(n-i,p+i) for n>=1. - Michael Wallner, Apr 20 2017
From Michael Wallner, Jan 31 2022: (Start)
a(n) = r(n,n) where r(n,m)=(m+1)*r(n-1,m)+r(n,m-1) for n>=m>=1, r(n,m)=0 for n=0.
a(n) = Theta(n!*4^n*exp(3*a1*n^(1/3))*n) for large n, where a1=-2.338... is the largest root of the Airy function Ai(x) of the first kind; see [Elvey Price, Fang, Wallner 2021]. (End)

A094216 Triangle read by rows giving the coefficients of formulas generating each variety of S1(n,k) (unsigned Stirling numbers of first kind). The p-th row (p>=1) contains T(i,p) for i=1 to 2p, where T(i,p) satisfies Sum_{i=1..2p} T(i,p) * C(n,i).

Original entry on oeis.org

1, 1, 2, 7, 8, 3, 6, 38, 93, 111, 65, 15, 24, 226, 874, 1821, 2224, 1600, 630, 105, 120, 1524, 8200, 24860, 47185, 58465, 47474, 24430, 7245, 945, 720, 11628, 81080, 326712, 852690, 1522375, 1905168, 1676325, 1018682, 407925, 97020, 10395, 5040

Offset: 1

André F. Labossière, May 27 2004, Feb 21 2007

The formulas S1(n+p,n) obtained are those of S1(n+2,n) { A000914 }, S1(n+3,n) { A001303 }, S1(n+4,n) { A000915 }, S1(n+5,n) { A053567 } and so on.

			Row 5 contains 120,1524,8200,24860,47185,58465,47474,24430,7245,945, so the formula generating S1(n+5,n) numbers { A053567 } will be the following : 120*n +1524*C(n,2) +8200*C(n,3) +24860*C(n,4) +47185*C(n,5) +58465*C(n,6) +47474*C(n,7) +24430*C(n,8) +7245*C(n,9) +945*C(n,10). And then substituting for the 10th number of such a S1(n+p,n) gives S1(15,10) = 37312275.

Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964, 9th Printing (1970), pp. 833-834.

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].
Francis L. Miksa (1901-1975), Stirling numbers of the first kind, "27 leaves reproduced from typewritten manuscript on deposit in the UMT File", Mathematical Tables and Other Aids to Computation, vol. 10, no. 53, January 1956, pp. 37-38 (Reviews and Descriptions of Tables and Books, 7[I]).
Dragoslav S. Mitrinovic (1908-1995), Sur les nombres de Stirling de première espèce et les polynômes de Stirling, AMS 11B73_05A19, Publications de la Faculté d'Electrotechnique de l'Université de Belgrade, Série Mathématiques et Physique (ISSN 0522-8441), no. 23, 1959 (5.V.1959), pp. 1-20.
John J. O'Connor and Edmund F. Robertson, James Stirling (1692-1770), (September 1998).
Eric Weisstein's World of Mathematics, Stirling numbers of the first kind.
Stephen Wolfram, Wolfram Research, Mathematica 5.2, webMathematica 2.

Cf. A000914, A001303, A000915, A053567, A008275, A008276.

Cf. A000012, A000217, A001147, A000142, A094262.

row[m_] := Module[{eq, t}, eq[n_] := Array[t, 2 m].Table[Binomial[n, k], {k, 1, 2 m}] == Abs[StirlingS1[n + m, n]]; Array[t, 2 m] /. Solve[ Array[ eq, 2 m]] // First];
Array[row, 7] // Flatten (* Jean-François Alcover, Nov 14 2019 *)

a(1,k) = k!
...
a(2*k-5,k) = a(2*k,k) * (175000*k^8 -2117500*k^7 +10856650*k^6 -30743377*k^5 +52511770*k^4 -55386931*k^3 +35321832*k^2 -12560580*k+1944000) / (1632960*k^3 -7348320*k^2 +9389520*k -3061800).
a(2*k-4,k) = a(2*k,k) * (2500*k^6 -17400*k^5 +48511*k^4 -69378*k^3 +53929*k^2 -21906*k +3744) / (7776*k^2-15552*k+5832).
a(2*k-3,k) = a(2*k,k) * (1250*k^4-4225*k^3+5023*k^2-2600*k+528) / (1620*k-810).
a(2*k-2,k) = a(2*k,k) * (50*k^3-93*k^2+55*k-12) / (36*k-18).
a(2*k-1,k) = a(2*k,k) * (5*k-2) / 3.
a(2*k,k) = (2*k)! / (k!*2^k).

A003703 Expansion of e.g.f. cos(log(1+x)).

Original entry on oeis.org

1, 0, -1, 3, -10, 40, -190, 1050, -6620, 46800, -365300, 3103100, -28269800, 271627200, -2691559000, 26495469000, -238131478000, 1394099824000, 15194495654000, -936096296850000, 29697351895900000, -819329864480400000, 21683886333440500000, -570263312237604700000

Offset: 0

R. H. Hardin, Simon Plouffe

			1 - x^2 + 3*x^3 - 10*x^4 + 40*x^5 - 190*x^6 + 1050*x^7 - 6620*x^8 + ...

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

Vaclav Kotesovec, Table of n, a(n) for n = 0..400 (first 100 terms from T. D. Noe)
Vaclav Kotesovec, Graph a(n+1)/a(n)
Vladimir Victorovich Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.
Eric Weisstein's World of Mathematics, Pochhammer Symbol.

Cf. A009024, A009454.

a:= n-> add(Stirling1(n, 2*k) * (-1)^(k), k=0..floor(n/2)):
seq(a(n), n=0..20);

CoefficientList[Series[Cos[Log[1 + x]], {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Feb 16 2015 *)
Table[(-1)^n Im[Pochhammer[1-I, n-1]], {n, 0, 20}] (* Vladimir Reshetnikov, Sep 13 2016 *)

{a(n) = if( n<0, 0, n! * polcoeff( cos( log( 1 + x + x * O(x^n))), n))} /* Michael Somos, Jul 26 2012 */

a(n) = (-1)^n*(prod(k=0, n-1, I+k)+prod(k=0, n-1, -I+k))/2; \\ Seiichi Manyama, Oct 10 2022

from sympy.functions.combinatorial.numbers import stirling
def A003703(n): return sum(stirling(n,k<<1,kind=1,signed=True)*(-1 if k&1 else 1) for k in range((n>>1)+1)) # Chai Wah Wu, Feb 22 2024

a(n) = Sum_{k=0..n-1} (-1)^(k+1)*T(n-k, k)*sin(Pi*(n-k-1)/2) + 0^n; T(n, k)=abs(A008276(n, k)). - Paul Barry, Apr 18 2005
abs(a(n)) = abs(f(n)) with f(n)=Product_{k=1..n} i+k (where i^2=-1). - Yalcin Aktar, Jul 13 2009
a(n) = Sum_{k=0..floor(n/2)} Stirling1(n,2*k)*(-1)^k. - Vladimir Kruchinin, Jan 29 2011
a(n+2)= -a(n+1)*(2*n+1) - a(n)*(1+n^2), a(0)=1, a(1)=0. - Sergei N. Gladkovskii, Aug 17 2012
a(n) = (-1)^n * ( (i)n + (-i)_n )/2, where (x)_n is the Pochhammer symbol and i is the imaginary unit. - _Seiichi Manyama, Oct 10 2022
a(n) = Re(gamma(i+1)/gamma(i+1-n)). The imaginary part is A009454. - Colin Beveridge, Jul 30 2024

A009454 Expansion of e.g.f. sin(log(1+x)).

Original entry on oeis.org

0, 1, -1, 1, 0, -10, 90, -730, 6160, -55900, 549900, -5864300, 67610400, -839594600, 11186357000, -159300557000, 2416003824000, -38894192662000, 662595375078000, -11911522255750000, 225382826562400000

Offset: 0

R. H. Hardin

Cf. A003703, A242652, A231530.

CoefficientList[Series[Sin[Log[1+x]], {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Jan 24 2015 *)
FullSimplify[Table[-((-1)^n*(Gamma[1 + I]*Gamma[-I + n] + Gamma[1 - I]*Gamma[I + n])*Sinh[Pi]) / (2*Pi), {n, 0, 20}]] (* Vaclav Kotesovec, Jan 24 2015 *)
Table[-(-1)^n Re[Pochhammer[1+I, n-1]], {n, 0, 20}] (* Vladimir Reshetnikov, Sep 13 2016 *)

sum(stirling1(n,2*k+1)*(-1)^(k),k,0,n/2) /* Vladimir Kruchinin, Aug 03 2010 */

from sympy.functions.combinatorial.numbers import stirling
def A009454(n): return sum(stirling(n,(k<<1)+1,kind=1,signed=True)*(-1 if k&1 else 1) for k in range(n+1>>1)) # Chai Wah Wu, Feb 22 2024

a(n) = Sum_{k=0..n-1} (-1)^k*T(n-1, k)*cos(Pi*(n-k-1)/2); T(n, k) = abs(A008276(n, k)). - Paul Barry, Apr 18 2005
abs(a(n)) = abs(Re(Product_{k=1..n-1} (k+I))) with I^2 = -1. - Yalcin Aktar, Jul 02 2005
a(n+2) = -(2n+1)*a(n+1)-(n^2+1)*a(n), a(0)=0, a(1)=1. - Remy Lachaud (pacifik31(AT)aol.com), Dec 25 2005
a(n) = Sum_{k=0..n/2} Stirling1(n,2k+1)*(-1)^k. - Vladimir Kruchinin, Aug 03 2010
a(n) = Im(gamma(i+1)/gamma(i+1-n)). The real part is A003703. - Colin Beveridge, Jul 30 2024

Extended with signs by Olivier Gérard, Mar 15 1997

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.

Original entry on oeis.org

-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

A054654 Triangle of Stirling numbers of 1st kind, S(n, n-k), n >= 0, 0 <= k <= n.

Original entry on oeis.org

1, 1, 0, 1, -1, 0, 1, -3, 2, 0, 1, -6, 11, -6, 0, 1, -10, 35, -50, 24, 0, 1, -15, 85, -225, 274, -120, 0, 1, -21, 175, -735, 1624, -1764, 720, 0, 1, -28, 322, -1960, 6769, -13132, 13068, -5040, 0

Offset: 0

N. J. A. Sloane, Apr 18 2000

Triangle is the matrix product of the binomial coefficients with the Stirling numbers of the first kind.
Triangle T(n,k) giving coefficients in expansion of n!*C(x,n) in powers of x. E.g., 3!*C(x,3) = x^3-3*x^2+2*x.
The matrix product of binomial coefficients with the Stirling numbers of the first kind results in the Stirling numbers of the first kind again, but the triangle is shifted by (1,1).
Essentially [1,0,1,0,1,0,1,0,...] DELTA [0,-1,-1,-2,-2,-3,-3,-4,-4,...] where DELTA is the operator defined in A084938; mirror image of the Stirling-1 triangle A048994. - Philippe Deléham, Dec 30 2006
From Doudou Kisabaka, Dec 18 2009: (Start)
The sum of the entries on each row of the triangle, starting on the 3rd row, equals 0. E.g., 1+(-3)+2+0 = 0.
The entries on the triangle can be computed as follows. T(n,r) = T(n-1,r) - (n-1)*T(n-1,r-1). T(n,r) = 0 when r equals 0 or r > n. T(n,r) = 1 if n==1. (End)

			Matrix begins:
  1, 0,  0,  0,  0,   0,    0,     0,      0, ...
  0, 1, -1,  2, -6,  24, -120,   720,  -5040, ...
  0, 0,  1, -3, 11, -50,  274, -1764,  13068, ...
  0, 0,  0,  1, -6,  35, -225,  1624, -13132, ...
  0, 0,  0,  0,  1, -10,   85,  -735,   6769, ...
  0, 0,  0,  0,  0,   1,  -15,   175,  -1960, ...
  0, 0,  0,  0,  0,   0,    1,   -21,    322, ...
  0, 0,  0,  0,  0,   0,    0,     1,    -28, ...
  0, 0,  0,  0,  0,   0,    0,     0,      1, ...
  ...
Triangle begins:
  1;
  1,   0;
  1,  -1,   0;
  1,  -3,   2,    0;
  1,  -6,  11,   -6,    0;
  1, -10,  35,  -50,   24,     0;
  1, -15,  85, -225,  274,  -120,   0;
  1, -21, 175, -735, 1624, -1764, 720, 0;
  ...

Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 18, table 18:6:1 at page 152.

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
Eric Weisstein's World of Mathematics, Pochhammer Symbol.
Eric Weisstein's World of Mathematics, Rising Factorial.
Eric Weisstein's World of Mathematics, FallingFactorial.

Essentially Stirling numbers of first kind, multiplied by factorials - see A008276.
The Stirling2 counterpart is A106800.
Cf. A054655, A039810, A039814, A126350, A126351, A126353, A340556.

a054654 n k = a054654_tabl !! n !! k
a054654_row n = a054654_tabl !! n
a054654_tabl = map reverse a048994_tabl
-- Reinhard Zumkeller, Mar 18 2014

a054654_row := proc(n) local k; seq(coeff(expand((-1)^n*pochhammer (-x,n)),x,n-k),k=0..n) end: # Peter Luschny, Nov 28 2010
seq(seq(Stirling1(n, n-k), k=0..n), n=0..8); # Peter Luschny, Feb 21 2021

row[n_] := Reverse[ CoefficientList[ (-1)^n*Pochhammer[-x, n], x] ]; Flatten[ Table[ row[n], {n, 0, 8}]] (* Jean-François Alcover, Feb 16 2012, after Maple *)
Table[StirlingS1[n,n-k],{n,0,10},{k,0,n}]//Flatten (* Harvey P. Dale, Jun 17 2023 *)

PARI
```
T(n,k)=polcoeff(n!*binomial(x,n), n-k)
```

n!*binomial(x, n) = Sum_{k=0..n} T(n, k)*x^(n-k).
(In Maple notation:) Matrix product A*B of matrix A[i,j]:=binomial(j-1,i-1) with i = 1 to p+1, j = 1 to p+1, p=8 and of matrix B[i,j]:=stirling1(j,i) with i from 1 to d, j from 1 to d, d=9.
T(n, k) = (-1)^k*Sum_{j=0..k} E2(k, j)*binomial(n+j-1, 2*k), where E2(k, j) are the second-order Eulerian numbers A340556. - Peter Luschny, Feb 21 2021

Additional comments from Thomas Wieder, Dec 29 2006

Edited by N. J. A. Sloane at the suggestion of Eric W. Weisstein, Jan 20 2008

A126671 Triangle read by rows: row n (n>=0) has g.f. Sum_{i=1..n} n!x^i(1+x)^(n-i)/(n+1-i).

Original entry on oeis.org

0, 0, 1, 0, 1, 3, 0, 2, 7, 11, 0, 6, 26, 46, 50, 0, 24, 126, 274, 326, 274, 0, 120, 744, 1956, 2844, 2556, 1764, 0, 720, 5160, 16008, 28092, 30708, 22212, 13068, 0, 5040, 41040, 147120, 304464, 401136, 351504, 212976, 109584, 0, 40320

Offset: 1

N. J. A. Sloane and Carlo Wood (carlo(AT)alinoe.com), Feb 13 2007

The first nonzero column gives the factorial numbers, which are Stirling_1(*,1), the rightmost diagonal gives Stirling_1(*,2), so this triangle may be regarded as interpolating between the first two columns of the Stirling numbers of the first kind.
This is a slice (the right-hand wall) through the infinite square pyramid described in the link. The other three walls give A007318 and A008276 (twice).
The coefficients of the A165674 triangle are generated by the asymptotic expansion of the higher order exponential integral E(x,m=2,n). The a(n) formulas for the coefficients in the right hand columns of this triangle lead to Wiggen's triangle A028421 and their o.g.f.s. lead to the sequence given above. Some right hand columns of the A165674 triangle are A080663, A165676, A165677, A165678 and A165679. - Johannes W. Meijer, Oct 07 2009

			Triangle begins:
0,
0, 1,
0, 1, 3,
0, 2, 7, 11,
0, 6, 26, 46, 50,
0, 24, 126, 274, 326, 274,
0, 120, 744, 1956, 2844, 2556, 1764,
0, 720, 5160, 16008, 28092, 30708, 22212, 13068,
0, 5040, 41040, 147120, 304464, 401136, 351504, 212976, 109584,
0, 40320, 367920, 1498320, 3582000, 5562576, 5868144, 4292496, 2239344, 1026576, ...

N. J. A. Sloane, Notes on Carlo Wood's Polynomials

Columns give A000142, A108217, A126672; diagonals give A000254, A067318, A126673. Row sums give A126674. Alternating row sums give A000142.
See A126682 for the full pyramid of coefficients of the underlying polynomials.
Cf. A165674, A028421, A080663, A165676, A165677, A165678 and A165679. - Johannes W. Meijer, Oct 07 2009

for n from 1 to 15 do t1:=add( n!*x^i*(1+x)^(n-i)/(n+1-i), i=1..n); series(t1,x,100); lprint(seriestolist(%)); od:

Join[{{0}}, Reap[For[n = 1, n <= 15, n++, t1 = Sum[n!*x^i*(1+x)^(n-i)/(n+1-i), {i, 1, n}]; se = Series[t1, {x, 0, 100}]; Sow[CoefficientList[se, x]]]][[2, 1]]] // Flatten (* Jean-François Alcover, Jan 07 2014, after Maple *)

Recurrence: T(n,0) = 0; for n>=0, i>=1, T(n+1,i) = (n+1)*T(n,i) + n!*binomial(n,i).

E.g.f.: x*log(1-(1+x)*y)/(x*y-1)/(1+x). - Vladeta Jovovic, Feb 13 2007

A054655 Triangle T(n,k) giving coefficients in expansion of n!*binomial(x-n,n) in powers of x.

Original entry on oeis.org

1, 1, -1, 1, -5, 6, 1, -12, 47, -60, 1, -22, 179, -638, 840, 1, -35, 485, -3325, 11274, -15120, 1, -51, 1075, -11985, 74524, -245004, 332640, 1, -70, 2086, -34300, 336049, -1961470, 6314664, -8648640, 1, -92, 3682, -83720, 1182769

Offset: 0

N. J. A. Sloane, Apr 18 2000

			Triangle begins:
  1;
  1,  -1;
  1,  -5,    6;
  1, -12,   47,    -60;
  1, -22,  179,   -638,    840;
  1, -35,  485,  -3325,  11274,   -15120;
  1, -51, 1075, -11985,  74524,  -245004,  332640;
  1, -70, 2086, -34300, 336049, -1961470, 6314664, -8648640;
  ...

Robert Israel, Table of n, a(n) for n = 0..10010 (rows 0 to 140, flattened).
Takao Komatsu, Some explicit values of a q-multiple zeta function at roots of unity, arXiv:2505.09357 [math.NT], 2025. See p. 10.

Cf. A000407, A054649, A054651, A054654, A008276.

a054655_row := proc(n) local k; seq(coeff(expand((-1)^n*pochhammer (n-x,n)),x,n-k),k=0..n) end: # Peter Luschny, Nov 28 2010

row[n_] := Table[ Coefficient[(-1)^n*Pochhammer[n - x, n], x, n - k], {k, 0, n}]; A054655 = Flatten[ Table[ row[n], {n, 0, 8}]] (* Jean-François Alcover, Apr 06 2012, after Maple *)

T(n,k)=polcoef(n!*binomial(x-n,n), n-k);

n!*binomial(x-n, n) = Sum_{k=0..n} T(n, k)*x^(n-k).
From Robert Israel, Jul 12 2016: (Start)
G.f.: Sum_{n>=0} Sum_{k=0..n} T(n,k)*x^n*y^k  = hypergeom([1, -1/(2*y), (1/2)*(-1+y)/y], [-1/y], -4*x*y).
E.g.f.: Sum_{n>=0} Sum_{k=0..n} T(n,k)*x^n*y^k/n! = (1+4*x*y)^(-1/2)*((1+sqrt(1+4*x*y))/2)^(1+1/y). (End)

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cp's OEIS Frontend

A008278 Reflected triangle of Stirling numbers of 2nd kind, S(n,n-k+1), n >= 1, 1 <= k <= n.

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A082161 Number of deterministic completely defined initially connected acyclic automata with 2 inputs and n transient unlabeled states (and a unique absorbing state).

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A094216 Triangle read by rows giving the coefficients of formulas generating each variety of S1(n,k) (unsigned Stirling numbers of first kind). The p-th row (p>=1) contains T(i,p) for i=1 to 2*p, where T(i,p) satisfies Sum_{i=1..2*p} T(i,p) * C(n,i).

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A003703 Expansion of e.g.f. cos(log(1+x)).

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A009454 Expansion of e.g.f. sin(log(1+x)).

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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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A094216 Triangle read by rows giving the coefficients of formulas generating each variety of S1(n,k) (unsigned Stirling numbers of first kind). The p-th row (p>=1) contains T(i,p) for i=1 to 2p, where T(i,p) satisfies Sum_{i=1..2p} T(i,p) * C(n,i).

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.

A126671 Triangle read by rows: row n (n>=0) has g.f. Sum_{i=1..n} n!x^i(1+x)^(n-i)/(n+1-i).