A143491 -id:A143491 - cp's OEIS frontend

A136124 Triangle read by rows: T(n,k) = (-1)^(n+k)*Sum_{j=1..k} s(n,j), where s(n,j) are the signed Stirling numbers of the first kind (n >= 2; 1 <= k <= n-1; s(n,j) = A008275(n,j)).

1, 2, 1, 6, 5, 1, 24, 26, 9, 1, 120, 154, 71, 14, 1, 720, 1044, 580, 155, 20, 1, 5040, 8028, 5104, 1665, 295, 27, 1, 40320, 69264, 48860, 18424, 4025, 511, 35, 1, 362880, 663696, 509004, 214676, 54649, 8624, 826, 44, 1, 3628800, 6999840, 5753736, 2655764

Offset: 2

Emeric Deutsch, Dec 23 2007

Sum of entries in row n = n!/2 = A001710(n). T(n,1) = (n-1)! = A000142(n-1). Columns 2,3,4 and 5 yield A001705,A001706,A001707 and A001708, respectively.
See A143491 for the interpretation of these numbers as restricted Stirling numbers of the first kind. See A049444 for a signed version of this array. - Peter Bala, Aug 25 2008
With offset n=0, k=0: triangle T(n,k), read by rows, given by [2,1,3,2,4,3,5,4,6,5,...] DELTA [1,0,1,0,1,0,1,0,1,0,1,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Sep 29 2011
With offset n=0, k=0: T(n,k) is the number of ways to seat n people at any number of round tables and serve exactly k of the tables water, some number of the remaining tables red wine, and the rest of the tables white wine. - Geoffrey Critzer, Mar 13 2015

			T(6,3)=71 because (-1)^9*[s(6,1)+s(6,2)+s(6,3)]=-(-120+274-225)=71.
Triangle starts:
    1;
    2,   1;
    6,   5,   1;
   24,  26,   9,   1;
  120, 154,  71,  14,   1;

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Olivier Bodini, Antoine Genitrini, Mehdi Naima, Ranked Schröder Trees, arXiv:1808.08376 [cs.DS], 2018.

Cf. A000142, A008275, A001705, A001706, A001707, A001708, A001710.

Cf. A049444, A143491. [Peter Bala, Aug 25 2008]

A136124_row := proc(n) local k,j; `if`(n=0,1,seq((-1)^(n+1-k)*add(stirling1(n+1,j), j=1..k),k=1..n)) end: seq(print(A136124_row(r)),r=1..6); # Peter Luschny, Sep 29 2011
with(combinat): T:=proc(n, k) options operator, arrow: (-1)^(n+k)*(sum(stirling1(n,j),j=1..k)) end proc: for n from 2 to 11 do seq(T(n,k),k=1..n-1) end do; # yields sequence in triangular form

nn = 10; Map[Select[#, # > 0 &] &,Range[0,nn]!CoefficientList[Series[Exp[(2 + y) Log[1/(1 - x)]], {x, 0, nn}], {x,y}]] // Flatten (* Geoffrey Critzer, Mar 13 2015 *)

E.g.f.: Sum[(1/n!)T(n,k)x^n*t^k, k=1..n-1, n>=2]=1/[(1+t)(1-x)^t]-(1+tx)/(1+t). Generating polynomial of row n = t*Product(j+t, j=2..n-1). T(n,k) is the sum of all products of n-k-1 different integers taken from {2,3,...,n-1}. For example, T(6,3) = 2*3 + 2*4 + 2*5 + 3*4 + 3*5 + 4*5 = 71.

A269952 Triangle read by rows, T(n,k) = Sum_{j=0..n} (-1)^(n-j)C(-j,-n)S2(j,k), S2 the Stirling set numbers A048993, for n>=0 and 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 4, 5, 1, 0, 8, 19, 9, 1, 0, 16, 65, 55, 14, 1, 0, 32, 211, 285, 125, 20, 1, 0, 64, 665, 1351, 910, 245, 27, 1, 0, 128, 2059, 6069, 5901, 2380, 434, 35, 1, 0, 256, 6305, 26335, 35574, 20181, 5418, 714, 44, 1

Offset: 0

Peter Luschny, Apr 10 2016

			1,
0, 1,
0, 2, 1,
0, 4, 5, 1,
0, 8, 19, 9, 1,
0, 16, 65, 55, 14, 1,
0, 32, 211, 285, 125, 20, 1,
0, 64, 665, 1351, 910, 245, 27, 1.

Peter Luschny, Extensions of the binomial

Variant: A143494 (the main entry for this triangle).
A005493 (row sums), A074051 (alt. row sums), A000079 (col. 1), A001047 (col. 2),
A016269 (col. 3), A025211 (col. 4), A000096 (diag. n,n-1), A215862 (diag. n,n-2),
A049444, A136124, A143491 (matrix inverse).
Cf. A048993, A269951.

A269952 := (n,k) -> Stirling2(n+1, k+1) - Stirling2(n, k+1):
seq(seq(A269952(n,k), k=0..n), n=0..9);

Flatten[ Table[ Sum[(-1)^(n-j) Binomial[-j,-n] StirlingS2[j,k], {j,0,n}], {n,0,9}, {k,0,n}]]

T(n, k) = S2(n+1, k+1) - S2(n, k+1).

A347989 a(n) = [x^n] (2n)! Sum_{k=0..2*n} binomial(x+k,k).

Original entry on oeis.org

1, 5, 71, 1665, 54649, 2310945, 119753843, 7353403057, 522289211873, 42137920501677, 3807384320667135, 380929847762489025, 41811136672902061321, 4995760464106519955705, 645541681316043216096315, 89705032647088734873129825, 13340173206548155385625683265, 2114001534402053456524492822485

Offset: 0

Seiichi Manyama, Sep 23 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..325

Cf. A001706, A001707, A001708, A001709, A008275, A143491, A347987.

a(n) = (2*n)!*polcoef(sum(k=n, 2*n, binomial(x+k, k)), n);

a(n) = (2*n)!*sum(k=n, 2*n, (2*n+1-k)*abs(stirling(k, n, 1))/k!);

a(n) = (2*n)! * Sum_{k=n..2*n} (2*n+1-k) * |Stirling1(k,n)|/k!.
a(n) = [x^(2*n)] ((2*n)!/n!) * (-log(1 - x))^n/(1 - x)^2.
From Vaclav Kotesovec, Sep 23 2021, updated May 14 2025: (Start)
a(n) = [x^n] Gamma(2*n + x + 2) / Gamma(x + 2).
a(n) ~ c * d^n * (n-1)!, where d = 8*w^2/(2*w-1), w = -LambertW(-1,-exp(-1/2)/2) and c = w^2 * sqrt(2) / (sqrt(w-1)*Pi) = 1.5967712192197964362930380385801737624829174112909160160618... (End)

A295225 Sum of antidiagonals of triangle of 2-Stirling numbers of the first kind.

Original entry on oeis.org

0, 0, 1, 2, 7, 29, 147, 883, 6156, 48942, 437404, 4343041, 47444364, 565601104, 7307463941, 101713884677, 1517483769898, 24157610375637, 408751073418742, 7325240566502843, 138608292368966173, 2761503816421313729, 57782245813372891911, 1266895493728240876072

Offset: 2

Eric M. Schmidt, Nov 18 2017

nonn

Edyta Hetmaniok, Barbara Smoleń, Roman Wituła, The Stirling triangles, Proceedings of the Symposium for Young Scientists in Technology, Engineering and Mathematics (SYSTEM 2017), Kaunas, Lithuania, April 28, 2017, p. 35-41.

a(n) = Sum_{k=0..[n/2]} A143491(n-k,k).

cp's OEIS Frontend

A136124 Triangle read by rows: T(n,k) = (-1)^(n+k)*Sum_{j=1..k} s(n,j), where s(n,j) are the signed Stirling numbers of the first kind (n >= 2; 1 <= k <= n-1; s(n,j) = A008275(n,j)).

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A269952 Triangle read by rows, T(n,k) = Sum_{j=0..n} (-1)^(n-j)C(-j,-n)S2(j,k), S2 the Stirling set numbers A048993, for n>=0 and 0<=k<=n.

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A347989 a(n) = [x^n] (2n)! Sum_{k=0..2*n} binomial(x+k,k).

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A295225 Sum of antidiagonals of triangle of 2-Stirling numbers of the first kind.

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Author

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cp's OEIS Frontend

A136124 Triangle read by rows: T(n,k) = (-1)^(n+k)*Sum_{j=1..k} s(n,j), where s(n,j) are the signed Stirling numbers of the first kind (n >= 2; 1 <= k <= n-1; s(n,j) = A008275(n,j)).

Views

Author

Keywords

Comments

Examples

Links

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Programs

Maple

Mathematica

Formula

A269952 Triangle read by rows, T(n,k) = Sum_{j=0..n} (-1)^(n-j)*C(-j,-n)*S2(j,k), S2 the Stirling set numbers A048993, for n>=0 and 0<=k<=n.

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Author

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Examples

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Programs

Maple

Mathematica

Formula

A347989 a(n) = [x^n] (2*n)! * Sum_{k=0..2*n} binomial(x+k,k).

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Author

Keywords

Links

Crossrefs

Programs

PARI

PARI

Formula

A295225 Sum of antidiagonals of triangle of 2-Stirling numbers of the first kind.

Views

Author

Keywords

Links

Formula

A269952 Triangle read by rows, T(n,k) = Sum_{j=0..n} (-1)^(n-j)C(-j,-n)S2(j,k), S2 the Stirling set numbers A048993, for n>=0 and 0<=k<=n.

A347989 a(n) = [x^n] (2n)! Sum_{k=0..2*n} binomial(x+k,k).