cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-9 of 9 results.

A317274 a(n) = Sum_{k=0..n} binomial(n,k)*Stirling1(n,k).

Original entry on oeis.org

1, 1, -1, -2, 19, -79, 76, 2640, -36945, 329371, -1861949, -4438774, 355714228, -7292531180, 109844527612, -1277006731104, 8181112825231, 124379387459175, -6806984421310187, 191750786928500050, -4289244423048443149, 80163499107525756105, -1146313133241947091420, 5494990440819210736560
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 25 2018

Keywords

Comments

Binomial convolution of the signed Stirling numbers of the first kind.

Crossrefs

Cf. A008275, A048994, A122455, A211210, A211211.

Programs

  • Mathematica
    Table[Sum[Binomial[n, k] StirlingS1[n, k], {k, 0, n}], {n, 0, 23}]
  • PARI
    a(n) = sum(k=0, n, binomial(n, k)*stirling(n, k, 1)); \\ Michel Marcus, Aug 07 2019

A211211 sum( C(n+1,k)*|S1(n,k)|, k=0..n). Binomial convolution of the Stirling numbers of the first kind.

Original entry on oeis.org

1, 2, 6, 30, 205, 1750, 17766, 207942, 2746815, 40315858, 649688072, 11387466948, 215440517656, 4371810051908, 94649397546302, 2176321870192342, 52938365091640943, 1357592080006964806, 36593629200726397630, 1033979281229140895582, 30552322294916306960625
Offset: 0

Views

Author

Olivier Gérard, Oct 23 2012

Keywords

Crossrefs

Shifted version of A211210. S1 Analog of A134094.

Programs

  • Mathematica
    Table[Sum[Binomial[n + 1, k] Abs[StirlingS1[n, k]], {k, 0, n}], {n, 0, 20}]

A047798 a(n) = Sum_{k=0..n} C(n,k)*Stirling2(n,k)^2.

Original entry on oeis.org

1, 1, 3, 31, 443, 9006, 241147, 7956579, 318973867, 15061651528, 824029357046, 51526959899570, 3636995712432667, 287053182699020609, 25126145438688593769, 2421761360666327615911, 255466264644678162575691, 29336098320197429601856772
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A008277, A047799, A211210, A317274.

Programs

  • GAP
    List([0..20], n-> Sum([0..n], k-> Binomial(n,k)*Stirling2(n,k)^2 )); # G. C. Greubel, Aug 07 2019
  • Magma
    [(&+[Binomial(n,k)*StirlingSecond(n,k)^2: k in [0..n]]): n in [0..20]]; // G. C. Greubel, Aug 07 2019
  • Maple
    seq(add(binomial(n,k)*stirling2(n, k)^2, k = 0..n), n = 0..20); # G. C. Greubel, Aug 07 2019
  • Mathematica
    Table[Sum[Binomial[n, k]*StirlingS2[n, k]^2, {k,0,n}], {n,0,20}] (* G. C. Greubel, Aug 07 2019 *)
  • PARI
    {a(n) = sum(k=0,n, binomial(n,k)*stirling(n,k,2)^2)};
vector(20, n, n--; a(n)) \\ G. C. Greubel, Aug 07 2019
  • Sage
    [sum(binomial(n,k)*stirling_number2(n,k)^2 for k in (0..n)) for n in (0..20)] # G. C. Greubel, Aug 07 2019

A187555 Triangle read by rows, defined by T(n,k)=binomial(n,k)*|Stirling1(n,k)|, 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 6, 9, 1, 0, 24, 66, 24, 1, 0, 120, 500, 350, 50, 1, 0, 720, 4110, 4500, 1275, 90, 1, 0, 5040, 37044, 56840, 25725, 3675, 147, 1, 0, 40320, 365904, 735392, 473830, 109760, 9016, 224, 1, 0, 362880, 3945024, 9922416, 8477784, 2828574, 381024, 19656, 324, 1, 0, 3628800, 46195920, 140724000, 151972800, 67869900, 13287330, 1134000, 39150, 450, 1
Offset: 0

Views

Author

Emanuele Munarini, Mar 11 2011

Keywords

Examples

			Triangle begins:
1
0,1
0,2,1
0,6,9,1
0,24,66,24,1
0,120,500,350,50,1
0,720,4110,4500,1275,90,1
0,5040,37044,56840,25725,3675,147,1
0,40320,365904,735392,473830,109760,9016,224,1

Links

Crossrefs

Row sum sequence is A211210.

Programs

  • Maple
    seq(seq(binomial(n,k)*abs(combinat[stirling1](n,k)),k=0..n),n=0..8);
  • Mathematica
    Flatten[Table[
  Table[Binomial[n, k] Abs[StirlingS1[n, k]], {k, 0, n}], {n, 0, 10}], 1]
  • Maxima
    create_list(binomial(n,k)*abs(stirling1(n,k)),n,0,10,k,0,n);

Formula

a(n,k) = binomial(n,k)*A132393(n,k).

Extensions

Edited by Olivier Gérard, Oct 23 2012

A247329 a(n) = Sum_{k=0..n} (-1)^(n-k) * C(n,k) * Stirling1(n+1, k+1).

Original entry on oeis.org

1, 2, 9, 58, 475, 4666, 53116, 684762, 9833391, 155341258, 2673209561, 49717424868, 992847765988, 21172798741316, 479921234767976, 11516219861132586, 291523666535143823, 7761036379846481206, 216699016885046232187, 6330257697841339549706, 193043926318644060255531
Offset: 0

Views

Author

Paul D. Hanna, Sep 26 2014

Keywords

Examples

			Illustration of initial terms:
a(0) = 1*1 = 1 ;
a(1) = 1*1 + 1*1 = 2 ;
a(2) = 1*2 + 2*3 + 1*1 = 9 ;
a(3) = 1*6 + 3*11 + 3*6 + 1*1 = 58 ;
a(4) = 1*24 + 4*50 + 6*35 + 4*10 + 1*1 = 475 ;
a(5) = 1*120 + 5*274 + 10*225 + 10*85 + 5*15 + 1*1 = 4666 ;
a(6) = 1*720 + 6*1764 + 15*1624 + 20*735 + 15*175 + 6*21 + 1*1 = 53116 ;
a(7) = 1*5040 + 7*13068 + 21*13132 + 35*6769 + 35*1960 + 21*322 + 7*28 + 1*1 = 684762 ; ...

Links

Crossrefs

Cf. A008275 (Stirling1 numbers), A211210.

Programs

  • Maple
    f:= proc(n) local k; add((-1)^(n-k)*binomial(n,k)*Stirling1(n+1,k+1),k=0..n); end proc:
map(f, [$0..30]); # Robert Israel, Aug 01 2019
  • Mathematica
    Table[Sum[(-1)^(n-k) * Binomial[n,k] * StirlingS1[n+1, k+1],{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Sep 29 2014 *)
  • PARI
    {Stirling1(n, k)=if(n==0, 1, n!*polcoeff(binomial(x, n), k))}
{a(n)=sum(k=0, n, (-1)^(n-k)*binomial(n,k)*Stirling1(n+1, k+1))}
for(n=0,30,print1(a(n),", "))

A343841 a(n) = Sum{k=0..n} (-1)^(n-k)*binomial(n, k)*Stirling2(n, k).

Original entry on oeis.org

1, 1, -1, -5, 15, 56, -455, -237, 16947, -64220, -529494, 6833608, -8606015, -459331677, 4335744673, 6800310151, -518075832085, 4315086396640, 19931595013738, -812870258798156, 6648395876520816, 46852711038750520, -1752440325584024944, 15485712825845269456
Offset: 0

Views

Author

Peter Luschny, May 04 2021

Keywords

Crossrefs

Cf. A048993, A122455, A211210, A317274.

Programs

  • Maple
    a := n -> add((-1)^(n-k)*binomial(n, k)*Stirling2(n, k), k=0..n):
seq(a(n), n = 0..24);
  • Mathematica
    a[n_] := Sum[(-1)^(n - k) * Binomial[n, k] * StirlingS2[n, k], {k, 0, n}]; Array[a, 24, 0] (* Amiram Eldar, May 07 2021 *)
  • PARI
    a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n, k)*stirling(n, k, 2)); \\ Michel Marcus, May 07 2021

A387152 Array read by ascending antidiagonals: A(n, k) = Sum_{j=0..n} binomial(k, j)*|Stirling1(n, j)|.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 2, 3, 3, 1, 0, 6, 7, 6, 4, 1, 0, 24, 23, 16, 10, 5, 1, 0, 120, 98, 57, 30, 15, 6, 1, 0, 720, 514, 257, 115, 50, 21, 7, 1, 0, 5040, 3204, 1407, 546, 205, 77, 28, 8, 1, 0, 40320, 23148, 9076, 3109, 1021, 336, 112, 36, 9, 1
Offset: 0

Views

Author

Peter Luschny, Aug 27 2025

Keywords

Examples

			Array begins:
  [0]  1,     1,      1,      1,       1,       1,       1, ...
  [1]  0,     1,      2,      3,       4,       5,       6, ...
  [2]  0,     1,      3,      6,      10,      15,      21, ...
  [3]  0,     2,      7,     16,      30,      50,      77, ...
  [4]  0,     6,     23,     57,     115,     205,     336, ...
  [5]  0,    24,     98,    257,     546,    1021,    1750, ...
  [6]  0,   120,    514,   1407,    3109,    6030,   10696, ...
  [7]  0,   720,   3204,   9076,   20695,   41330,   75356, ...
  [8]  0,  5040,  23148,  67456,  157865,  323005,  602517, ...
  [9]  0, 40320, 190224, 567836, 1358564, 2837549, 5396650, ...

Crossrefs

Rows: A000012 [0], A001477 [1], A000217 [2], A005581 [3], A387204 [4].
Columns: A000007 [0], A000142 [shifted, 1], A387205 [2].
Contains A271700 in transpose.
Cf. A211210 (main diagonal), A130534.

Programs

  • Maple
    A := (n, k) -> add(binomial(k, j)*abs(Stirling1(n, j)), j = 0..n):
seq(seq(A(n-k, k), k = 0..n), n = 0..10);
# Expanding rows or columns:
RowSer := n -> series((1+x)^k*GAMMA(x + n)/GAMMA(x), x, 12):
Trow := n -> k -> coeff(RowSer(n), x, k):
ColSer := n -> series(orthopoly:-L(n, log(1 - x)), x, 12):
Tcol := k -> n -> n! * coeff(ColSer(k), x, n):
seq(lprint(seq(Trow(n)(k), k = 0..7)), n = 0..9);
seq(lprint(seq(Tcol(k)(n), n = 0..7)), k = 0..9);
  • Python
    from functools import cache
@cache
def T(n: int, k: int) -> int:
    if n == 0: return 1
    if k == 0: return 0
    return (n - 1) * T(n - 1, k) + T(n, k - 1) - (n - 2) * T(n - 1, k - 1)
for n in range(7): print([T(n, k) for k in range(7)])

Formula

T(n, k) = n! * [x^n] Laguerre(k, log(1 - x)).
From Natalia L. Skirrow, Aug 27 2025: (Start)
D-finite with T(n,k) = (n-1)*T(n-1,k)+T(n,k-1)-(n-2)*T(n-1,k-1).
O.g.f.: hypergeom([1,y/(1-y)],[],x)/(1-y).
Row o.g.f.: (y/(1-y))_n/(1-y), where (x)_n is the Pochhammer symbol/rising factorial.
Row o.g.f. is also 0^n + y/(1-y)^(n+1)*Prod_{j=1..n-2}(j+1-j*y).
E.g.f.: 1/((1-y)*(1-x)^(y/(1-y))).
Column e.g.f.: hypergeom([-k],[1],log(1-y)).
T(n,k) = [x^k] (1+x)^k*(x)_n.
(End)

A271700 Triangle read by rows, T(n,k) = Sum_{j=0..n} (-1)^(n-j)*C(-j-1,-n-1)*S1(k,j), S1 the Stirling cycle numbers A132393, for n>=0 and 0<=k<=n.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 3, 6, 16, 1, 4, 10, 30, 115, 1, 5, 15, 50, 205, 1021, 1, 6, 21, 77, 336, 1750, 10696, 1, 7, 28, 112, 518, 2814, 17766, 128472, 1, 8, 36, 156, 762, 4308, 28050, 207942, 1734447, 1, 9, 45, 210, 1080, 6342, 42528, 322860, 2746815, 25937683
Offset: 0

Views

Author

Peter Luschny, Apr 14 2016

Keywords

Examples

			Triangle starts:
[1]
[1, 1]
[1, 2, 3]
[1, 3, 6,  16]
[1, 4, 10, 30,  115]
[1, 5, 15, 50,  205, 1021]
[1, 6, 21, 77,  336, 1750, 10696]
[1, 7, 28, 112, 518, 2814, 17766, 128472]

Crossrefs

A000027 (col. 1), A000217, A161680 (col. 2), A005581 (col. 3), A211210 (diag. n,n), A211211 (diag. n,n-1).

Programs

  • Maple
    T := (n,k) -> add(abs(Stirling1(k,j))*binomial(-j-1,-n-1)*(-1)^(n-j),j=0..n);
seq(seq(T(n,k), k=0..n), n=0..9);
  • Mathematica
    Flatten[Table[Sum[(-1)^(n-j)Binomial[-j-1,-n-1] Abs[StirlingS1[k,j]],{j,0,n}], {n,0,9},{k,0,n}]]

A344053 a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*Stirling2(n, k)*k!.

Original entry on oeis.org

1, 1, 0, -9, -40, 125, 3444, 18571, -241872, -5796711, -24387220, 1132278191, 25132445832, 8850583573, -10681029498972, -214099676807085, 1643397436986464, 176719161389104817, 2976468247699317468, -71662294521163070153, -4638920054290748840520, -55645074852328083377619
Offset: 0

Views

Author

Peter Luschny, May 10 2021

Keywords

Comments

Inverse binomial convolution of the Fubini numbers (A131689).

Crossrefs

Cf. A000312, A131689, A122455, A343841, A317274, A211210, A219859.

Programs

  • Mathematica
    a[n_] := Sum[(-1)^(n - k) * Binomial[n, k] * StirlingS2[n, k] * k!, {k, 0, n}]; Array[a, 22, 0] (* Amiram Eldar, May 10 2021 *)
  • PARI
    a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n, k)*stirling(n, k, 2)*k!); \\ Michel Marcus, May 10 2021

Formula

a(n) = Sum_{k=0..n} (-1)^k * A219859(n,k). - Alois P. Heinz, Jan 24 2022
a(n) = n! * [x^n] (2 - exp(-x))^n. - Fabian Pereyra, Aug 31 2024
Showing 1-9 of 9 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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