A094816 -id:A094816 - cp's OEIS frontend

A095722 E.g.f.: exp(x)/(1-x)^8.

1, 9, 89, 961, 11265, 142601, 1940089, 28245729, 438351041, 7226001865, 126122874201, 2324074591169, 45094140207169, 919088049256521, 19633713260950265, 438708172312264801, 10234490436580101249

Offset: 0

Philippe Deléham, Jul 08 2004

nonn

Sum_{k = 0..n} A094816(n,k)*x^k gives A000522(n), A001339(n), A082030(n), A095000(n), A095177(n), A096307(n), A096341(n) for x = 1, 2, 3, 4, 5, 6, 7 respectively.

G. C. Greubel, Table of n, a(n) for n = 0..250

With[{nn=20},CoefficientList[Series[Exp[x]/(1-x)^8,{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, May 26 2013 *)
Table[HypergeometricPFQ[{8, -n}, {}, -1], {n, 0, 20}] (* Benedict W. J. Irwin, May 27 2016 *)

a(n) = Sum_{k = 0..n} A094816(n, k)*8^k.
a(n) = Sum_{k = 0..n} binomial(n, k)*(k+7)! / 7!.
a(n) = 2F0(8,-n;;-1). - Benedict W. J. Irwin, May 27 2016

A381021 Expansion of e.g.f. log(1-x)^2 * exp(x) / 2.

Original entry on oeis.org

0, 0, 1, 6, 29, 145, 814, 5243, 38618, 321690, 2995011, 30840304, 348114711, 4274888891, 56744495872, 809667333733, 12358833406580, 200955441549140, 3467781770502885, 63298198354605210, 1218507112218768721, 24671782054230662277, 524152965820457130290

Offset: 0

Seiichi Manyama, Feb 12 2025

nonn

Column k=2 of A094816.

Cf. A073596.

nmax=22; CoefficientList[Series[Log[1-x]^2*Exp[x]/2,{x,0,nmax}],x]Range[0,nmax]! (* Stefano Spezia, Feb 12 2025 *)

a(n) = sum(k=0, n, binomial(n, k)*abs(stirling(k, 2, 1)));

a(n) = Sum_{k=0..n} binomial(n,k) * |Stirling1(k,2)|.

A095740 E.g.f.: exp(x)/(1-x)^9.

Original entry on oeis.org

1, 10, 109, 1288, 16417, 224686, 3288205, 51263164, 848456353, 14862109042, 274743964621, 5346258202000, 109249238631169, 2339328151461718, 52384307381414317, 1224472783033479556, 29826054965115774145

Offset: 0

Philippe Deléham Jul 09 2004

nonn

Sum_{k = 0..n} A094816(n,k)*x^k gives A000522(n), A001339(n), A082030(n), A095000(n), A095177(n), A096307(n), A096341(n), A095722(n) for x = 1, 2, 3, 4, 5, 6, 7, 8.

Robert Israel, Table of n, a(n) for n = 0..440

seq(simplify(hypergeom([9,-n],[],-1)),n=0..30); # Robert Israel, May 27 2016

Table[HypergeometricPFQ[{9, -n}, {}, -1], {n, 0, 20}] (* Benedict W. J. Irwin, May 27 2016 *)

a(n) = Sum_{k = 0..n} A094816(n, k)*9^k.
a(n) = Sum_{k = 0..n} binomial(n, k)*(k+8)!/8!.
a(n) = 2F0(9,-n;;-1). - Benedict W. J. Irwin, May 27 2016
a(n) = ((n^8 + 28*n^7 + 350*n^6 + 2492*n^5 + 10899*n^4 + 29596*n^3 + 48082*n^2 + 42048*n + 14833) * Gamma(n+1,1)*e + n^7 + 28*n^6 + 349*n^5 + 2465*n^4 + 10579*n^3 + 27501*n^2 + 40132*n + 25487) / 40320. - Robert Israel, May 27 2016

A381023 Expansion of e.g.f. log(1-x)^4 * exp(x) / 24.

Original entry on oeis.org

0, 0, 0, 0, 1, 15, 160, 1575, 15659, 163191, 1809905, 21474255, 272757166, 3703523824, 53631736795, 826097224680, 13497286183354, 233291225507890, 4254733292942982, 81680724157089634, 1646873959921840191, 34800264421134754997, 769198023696181428250, 17751664780107823096301

Offset: 0

Seiichi Manyama, Feb 12 2025

nonn

Column k=4 of A094816.

Cf. A381025.

nmax=23; CoefficientList[Series[Log[1-x]^4*Exp[x]/24, {x, 0, nmax}], x]Range[0, nmax]! (* Stefano Spezia, Feb 12 2025 *)

a(n) = sum(k=0, n, binomial(n, k)*abs(stirling(k, 4, 1)));

a(n) = Sum_{k=0..n} binomial(n,k) * |Stirling1(k,4)|.

A346844 E.g.f.: exp(exp(x) - 1) * (exp(x) - 1)^5 / 5!.

Original entry on oeis.org

1, 21, 287, 3290, 34671, 350889, 3492511, 34669734, 346231886, 3497726232, 35872743270, 374387203190, 3982122624117, 43207791878715, 478532965417529, 5411213661200830, 62482405993313229, 736696756305382411, 8868148033487285103, 108969560832001750716

Offset: 5

Ilya Gutkovskiy, Aug 05 2021

nonn

Cf. A000110, A000389, A000481, A003128, A005493, A049020, A094816, A346842, A346843.

b:= proc(n, m) option remember;
     `if`(n=0, binomial(m, 5), m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=5..24);  # Alois P. Heinz, Aug 05 2021

nmax = 24; CoefficientList[Series[Exp[Exp[x] - 1] (Exp[x] - 1)^5/5!, {x, 0, nmax}], x] Range[0, nmax]! // Drop[#, 5] &
Table[Sum[StirlingS2[n, k] Binomial[k, 5], {k, 0, n}], {n, 5, 24}]
Table[Sum[Binomial[n, k] StirlingS2[k, 5] BellB[n - k], {k, 0, n}], {n, 5, 24}]
Table[(-BellB[n] + 89*BellB[n+1] - 145*BellB[n+2] + 75*BellB[n+3] - 15*BellB[n+4] + BellB[n+5])/120, {n, 5, 24}] (* Vaclav Kotesovec, Aug 06 2021 *)

my(x='x+O('x^25)); Vec(serlaplace(exp(exp(x)-1)*(exp(x)-1)^5/5!)) \\ Michel Marcus, Aug 06 2021

a(n) = Sum_{k=0..n} Stirling2(n,k) * binomial(k,5).
a(n) = Sum_{k=0..n} binomial(n,k) * Stirling2(k,5) * Bell(n-k).
a(n) = (-Bell(n) + 89*Bell(n+1) - 145*Bell(n+2) + 75*Bell(n+3) - 15*Bell(n+4) + Bell(n+5))/120. - Vaclav Kotesovec, Aug 06 2021

A381082 Triangle T(n,k) read by rows, where the columns are the coefficients of the standard expansion of the function f(x) = (-log(1-x))^(k)exp(-mx)/k! for the case m=2.

Original entry on oeis.org

1, -2, 1, 4, -3, 1, -8, 8, -3, 1, 16, -18, 11, -2, 1, -32, 44, -20, 15, 0, 1, 64, -80, 94, 5, 25, 3, 1, -128, 272, 56, 294, 105, 49, 7, 1, 256, 112, 1868, 1596, 1169, 392, 98, 12, 1, -512, 5280, 12216, 16148, 10290, 4305, 1092, 186, 18, 1

Offset: 0

Igor Victorovich Statsenko, Feb 13 2025

			Triangle starts:
  [0]     1;
  [1]    -2,      1;
  [2]     4,     -3,       1;
  [3]    -8,      8,      -3,       1;
  [4]    16,    -18,      11,      -2,       1;
  [5]   -32,     44,     -20,      15,       0,        1;
  [6]    64,    -80,      94,       5,      25,        3,     1;
  [7]  -128,    272,      56,     294,     105,       49,     7,     1;
  [8]   256,    112,    1868,    1596,    1169,      392,    98,    12,    1;
  [9]  -512,   5280,   12216,   16148,   10290,     4305,  1092,   186,   18,     1;
  ...

Cf. A000023 (row sums).
Columns 0,1: A122803, A346397.
Triangles: for m = -3 is A327997; for m = -2 is A137346 (unsigned); for m = -1 is A094816; for m = 0 is A132393; for m = 1 is A269953.

T:=(m,n,k)->add(Stirling1(n-i,k)*binomial(n,i)*m^(i)*(-1)^(n-k), i=0..n):
m:=2:seq(print(seq(T(m,n,k), k=0..n)), n=0..9);

T(n,k) = Sum_{i=0..n} Stirling1(n-i, k)*binomial(n, i)*m^(i)*(-1)^(n-k), where m = 2.

A129334 Triangle T(n,k) read by rows: inverse of the matrix PE = exp(P)/exp(1) given in A011971.

Original entry on oeis.org

1, -1, 1, 0, -2, 1, 1, 0, -3, 1, 1, 4, 0, -4, 1, -2, 5, 10, 0, -5, 1, -9, -12, 15, 20, 0, -6, 1, -9, -63, -42, 35, 35, 0, -7, 1, 50, -72, -252, -112, 70, 56, 0, -8, 1, 267, 450, -324, -756, -252, 126, 84, 0, -9, 1, 413, 2670, 2250, -1080, -1890, -504, 210, 120, 0, -10, 1

Offset: 0

Gottfried Helms, Apr 08 2007

The structure of the triangle is A(r,c) = A000587(1+(r-c))*binomial(r-1,c-1) where row index r and column-index c start at 1.

Row polynomials defined recursively: P(0,x) = 1, P(n+1,x) = x*P(n,x) - P(n,x+1). The polynomials appear to be irreducible. Polynomials evaluated at x = c give sequences with e.g.f. exp(1 - cx - exp(-x)).

			Triangle starts:
[0]   1;
[1]  -1,   1;
[2]   0,  -2,    1;
[3]   1,   0,   -3,    1;
[4]   1,   4,    0,   -4,    1;
[5]  -2,   5,   10,    0,   -5,   1;
[6]  -9, -12,   15,   20,    0,  -6,  1;
[7]  -9, -63,  -42,   35,   35,   0, -7,  1;
[8]  50, -72, -252, -112,   70,  56,  0, -8,  1;
[9] 267, 450, -324, -756, -252, 126, 84,  0, -9, 1;

S. de Wannemacker, T. Laffey and R. Osburn, On a conjecture of Wilf, arXiv:math/0608085 [math.NT], 2006-2007.

First column is A000587 (Uppuluri Carpenter numbers) which is also the negative of the row sums (=P(n, 1)). Polynomials evaluated at 2 are A074051, at -1 A109747.

Cf. A094816.

P := proc(n,x) option remember; if n=0 then 1 else
x*P(n-1, x) - P(n-1, x+1) fi end:
aRow := n -> seq(coeff(P(n, x), x, k), k = 0..n):
seq(aRow(n), n = 0..10); # Peter Luschny, Apr 15 2022

Let P be the lower-triangular Pascal-matrix, PE = exp(P-I) a matrix-exponential in exact integer arithmetic (or PE = lim exp(P)/exp(1) as limit of the exponential) then A = PE^-1 and a(n) = A(n, read sequentially). - Gottfried Helms, Apr 08 2007

T(n, k) = Sum_{j=0..n} (-1)^(j-k)*A094816(j, k)*Stirling2(n, j). - Mélika Tebni, Apr 15 2022

Edited by Ralf Stephan, May 12 2007

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A095722 E.g.f.: exp(x)/(1-x)^8.

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A381021 Expansion of e.g.f. log(1-x)^2 * exp(x) / 2.

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A095740 E.g.f.: exp(x)/(1-x)^9.

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A381023 Expansion of e.g.f. log(1-x)^4 * exp(x) / 24.

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A346844 E.g.f.: exp(exp(x) - 1) * (exp(x) - 1)^5 / 5!.

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A381082 Triangle T(n,k) read by rows, where the columns are the coefficients of the standard expansion of the function f(x) = (-log(1-x))^(k)*exp(-m*x)/k! for the case m=2.

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A129334 Triangle T(n,k) read by rows: inverse of the matrix PE = exp(P)/exp(1) given in A011971.

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A381082 Triangle T(n,k) read by rows, where the columns are the coefficients of the standard expansion of the function f(x) = (-log(1-x))^(k)exp(-mx)/k! for the case m=2.