A001707 -id:A001707 - cp's OEIS frontend

A163934 Triangle related to the asymptotic expansion of E(x,m=4,n).

1, 6, 4, 35, 40, 10, 225, 340, 150, 20, 1624, 2940, 1750, 420, 35, 13132, 27076, 19600, 6440, 980, 56, 118124, 269136, 224490, 90720, 19110, 2016, 84, 1172700, 2894720, 2693250, 1265460, 330750, 48720, 3780, 120

Offset: 1

Johannes W. Meijer, Aug 13 2009

The higher order exponential integrals E(x,m,n) are defined in A163931 while the general formula for their asymptotic expansion can be found in A163932.
We used the latter formula and the asymptotic expansion of E(x,m=3,n), see A163932, to determine that E(x,m=4,n) ~ (exp(-x)/x^4)*(1 - (6+4*n)/x + (35+40*n+ 10*n^2)/x^2 - (225+340*n+ 150*n^2+20*n^3)/x^3 + ... ). This formula leads to the triangle coefficients given above.
The asymptotic expansion leads for the values of n from one to five to known sequences, see the cross-references.
The numerators of the o.g.f.s. of the right hand columns of this triangle lead for z=1 to A000457, see A163939 for more information.
The first Maple program generates the sequence given above and the second program generates the asymptotic expansion of E(x,m=4,n).

			The first few rows of the triangle are:
1;
6, 4;
35, 40, 10;
225, 340, 150, 20;

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A163931 (E(x,m,n)), A163932 and A163939.
Cf. A048994 (Stirling1), A000454 (row sums).
A000399, 4*A000454, 10*A000482, 20*A001233, 35*A001234 equal the first five left hand columns.
A000292, A027777 and A163935 equal the first three right hand columns.
The asymptotic expansion leads to A000454 (n=1), A001707 (n=2), A001713 (n=3), A001718 (n=4) and A001723 (n=5).
Cf. A130534 (m=1), A028421 (m=2), A163932 (m=3).

with(combinat): A163934 := proc(n,m): (-1)^(n+m)* binomial(m+2, 3) *stirling1(n+2, m+2) end: seq(seq(A163934(n,m), m=1..n), n=1..8);
with(combinat): imax:=6; EA:=proc(x,m,n) local E, i; E:=0: for i from m-1 to imax+2 do E:=E + sum((-1)^(m+k+1)*binomial(k,m-1)*n^(k-m+1)* stirling1(i, k), k=m-1..i)/x^(i-m+1) od: E:= exp(-x)/x^(m)*E: return(E); end: EA(x,4,n);
# Maple programs revised by Johannes W. Meijer, Sep 11 2012

a[n_, m_] /; n >= 1 && 1 <= m <= n = (-1)^(n+m)*Binomial[m+2, 3] * StirlingS1[n+2, m+2]; Flatten[Table[a[n, m], {n, 1, 8}, {m, 1, n}]][[1 ;; 36]] (* Jean-François Alcover, Jun 01 2011, after formula *)

a(n,m) = (-1)^(n+m)*C(m+2,3)*stirling1(n+2,m+2) for n >= 1 and 1<= m <= n.

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)).

Original entry on oeis.org

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.

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)

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A163934 Triangle related to the asymptotic expansion of E(x,m=4,n).

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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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Maple

Mathematica

Formula

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

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cp's OEIS Frontend

A163934 Triangle related to the asymptotic expansion of E(x,m=4,n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

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

Crossrefs

Programs

Maple

Mathematica

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

PARI

Formula

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