A051561 -id:A051561 - cp's OEIS frontend

A163932 Triangle related to the asymptotic expansion of E(x,m=3,n).

1, 3, 3, 11, 18, 6, 50, 105, 60, 10, 274, 675, 510, 150, 15, 1764, 4872, 4410, 1750, 315, 21, 13068, 39396, 40614, 19600, 4830, 588, 28, 109584, 354372, 403704, 224490, 68040, 11466, 1008, 36, 1026576, 3518100, 4342080, 2693250, 949095, 198450

Offset: 1

Johannes W. Meijer & Nico Baken (n.h.g.baken(AT)tudelft.nl), Aug 13 2009, Oct 22 2009

The higher order exponential integrals E(x,m,n) are defined in A163931. The general formula for the asymptotic expansion E(x,m,n) ~ E(x,m-1,n+1)/x - n*E(x,m-1,n+2)/x^2 + n*(n+1) * E(x,m-1,n+3)/x^3 - n*(n+1)*(n+2)*E(x,m-1,n+4)/x^4 + ...., m >= 1 and n >= 1.
We used this formula and the asymptotic expansion of E(x,m=2,n), see A028421, to determine that E (x,m=3,n) ~ (exp(-x)/x^3)*(1 - (3+3*n)/x + (11+18*n+6*n^2)/x^2 - (50+105*n+ 60*n^2+ 10*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 ten 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 A001879, see A163938 for more information.
The first Maple program generates the sequence given above and the second program generates the asymptotic expansion of E(x,m=3,n).

			The first few rows of the triangle are:
[1]
[3, 3]
[11, 18, 6]
[50, 105, 60, 10]

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

Cf. A163931 (E(x,m,n)) and A163938.
Cf. A048994 (Stirling1), A000399 (row sums).
A000254, 3*A000399, 6*A000454, 10*A000482, 15*A001233, 21*A001234 equal the first six left hand columns.
A000217, A006011 and A163933 equal the first three right hand columns.
The asymptotic expansion leads to A000399 (n=1), A001706 (n=2), A001712 (n=3), A001717 (n=4), A001722 (n=5), A051525 (n=6), A051546 (n=7), A051561 (n=8), A051563 (n=9) and A051565 (n=10).
Cf. A130534 (m=1), A028421 (m=2) and A163934 (m=4).

nmax:=8; with(combinat): for n1 from 1 to nmax do for m from 1 to n1 do a(n1, m) := (-1)^(n1+m)*binomial(m+1, 2)*stirling1(n1+1, m+1) od: od: seq(seq(a(n1,m), m=1..n1), n1=1..nmax);
# End program 1
with(combinat): imax:=6; EA:=proc(x, m, n) local E, i; E := 0: for i from m-1 to imax+1 do E := E + sum((-1)^(m+k1+1)*binomial(k1, m-1)*n^(k1-m+1)* stirling1(i, k1), k1=m-1..i)/x^(i-m+1) od: E := exp(-x)/x^(m)*E: return(E); end: EA(x, 3, n);
# End program 2

a[n_, m_] /; n >= 1 && 1 <= m <= n = (-1)^(n+m)*Binomial[m+1, 2] * StirlingS1[n+1, m+1]; Flatten[Table[a[n, m], {n, 1, 9}, {m, 1, n}]][[1 ;; 42]] (* Jean-François Alcover, Jun 01 2011, after formula *)

for(n=1,10, for(m=1,n, print1((-1)^(n+m)*binomial(m+1,2) *stirling(n+1,m+1,1), ", "))) \\ G. C. Greubel, Aug 08 2017

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

Edited by Johannes W. Meijer, Sep 22 2012

A376634 Triangle read by rows: T(n, k) = Sum_{i=0..n-k} Stirling1(i + m, m)binomial(n+m+1, n-k-i)(n + m - k)!/(i + m)!, for m = 2.

Original entry on oeis.org

1, 9, 1, 71, 12, 1, 580, 119, 15, 1, 5104, 1175, 179, 18, 1, 48860, 12154, 2070, 251, 21, 1, 509004, 133938, 24574, 3325, 335, 24, 1, 5753736, 1580508, 305956, 44524, 5000, 431, 27, 1, 70290936, 19978308, 4028156, 617624, 74524, 7155, 539, 30, 1, 924118272, 270074016, 56231712, 8969148, 1139292, 117454, 9850, 659, 33, 1, 13020978816, 3894932448, 832391136, 136954044, 18083484, 1961470, 176554, 13145, 791, 36, 1

Offset: 0

Igor Victorovich Statsenko, Sep 30 2024

The columns of the triangle T(m,n,k) represent the coefficients of the asymptotic expansion of the higher order exponential integral E(x,m+1,k+2), for m=2, k>=0. For reference see. A163931.

			Triangle starts:
 [0]          1;
 [1]          9,          1;
 [2]         71,         12,          1;
 [3]        580,        119,         15,        1;
 [4]       5104,       1175,        179,       18,        1;
 [5]      48860,      12154,       2070,      251,       21,       1;
 [6]     509004,     133938,      24574,     3325,      335,      24,     1;
 [7]    5753736,    1580508,     305956,    44524,     5000,     431,    27,     1;

Igor Victorovich Statsenko, Relationships of P-generalized Stirling numbers of the first kind with other generalized Stirling numbers, Innovation science No 10-1, State Ufa, Aeterna Publishing House, 2024, pp. 19-22. In Russian.

Column k: A001706 (k=0), A001712 (k=1), A001717 (k=2), A001722 (k=3), A051525 (k=4), A051546 (k=5), A051561 (k=6).

Cf. A094587 and A173333 for m=0, A376582 for m=1.

T:=(m,n,k)->add(Stirling1(i+m,m)*binomial(n+m+1,n-k-i)*(n+m-k)!/(i+m)!,i=0..n-k):m:=2:seq(seq(T(m,n,k), k=0..n),n=0..10);

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A163932 Triangle related to the asymptotic expansion of E(x,m=3,n).

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A376634 Triangle read by rows: T(n, k) = Sum_{i=0..n-k} Stirling1(i + m, m)binomial(n+m+1, n-k-i)(n + m - k)!/(i + m)!, for m = 2.

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cp's OEIS Frontend

A163932 Triangle related to the asymptotic expansion of E(x,m=3,n).

Views

Author

Keywords

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Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A376634 Triangle read by rows: T(n, k) = Sum_{i=0..n-k} Stirling1(i + m, m)*binomial(n+m+1, n-k-i)*(n + m - k)!/(i + m)!, for m = 2.

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Author

Keywords

Comments

Examples

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Programs

Maple

A376634 Triangle read by rows: T(n, k) = Sum_{i=0..n-k} Stirling1(i + m, m)binomial(n+m+1, n-k-i)(n + m - k)!/(i + m)!, for m = 2.