Nico Baken - cp's OEIS frontend

A161743 Fourth left hand column of the RSEG2 triangle A161739.

1, 10, 73, 425, 1561, -2856, -73520, 380160, 15376416, -117209664, -7506967104, 72162155520, 7045087741056, -80246202992640, -11448278791372800, 149576169325363200, 30017051616972275712, -440857664887810867200

Offset: 3

Johannes W. Meijer & Nico Baken (n.h.g.baken(AT)tudelft.nl), Jun 18 2009

Equals fourth left hand column of A161739 (RSEG2 triangle).
Other left hand columns are A129825 and A161742.
A008955 is a central factorial number triangle.
A028246 is Worpitzky's triangle.
A001710 (n!/2!), A001715 (n!/3!), A001720 (n!/4!), A001725 (n!/5!), A001730 (n!/6!), A049388 (n!/7!), A049389 (n!/8!), A049398 (n!/9!), A051431 (n!/10!) appear in Maple program.

nmax:=21; for n from 0 to nmax do A008955(n,0):=1 end do: for n from 0 to nmax do A008955(n,n):=(n!)^2 end do: for n from 1 to nmax do for m from 1 to n-1 do A008955(n,m):= A008955(n-1,m-1)*n^2+A008955(n-1,m) end do: end do: for n from 1 to nmax do A028246(n,1):=1 od: for n from 1 to nmax do A028246(n,n):=(n-1)! od: for n from 3 to nmax do for m from 2 to n-1 do A028246(n,m):=m*A028246(n-1,m)+(m-1)*A028246(n-1,m-1) od: od: for n from 3 to nmax do a(n) := sum(((-1)^k/((k+2)!*(k+3)!))*(n!)*A028246(n,k+3)* A008955(k+2,k), k=0..n-3) od: seq(a(n),n=3..nmax);

a(n) = sum(((-1)^k/((k+2)!*(k+3)!))*(n!)*A028246(n, k+3)*A008955(k+2, k), k = 0..n-3).

A161741 Fifth right hand column of the RSEG2 triangle A161739.

Original entry on oeis.org

-4, -14, 1561, 25809, 218769, 1284261, 5885671, 22482031, 74581507, 220977757, 596666070, 1490430630, 3484310310, 7693206894, 16160890914, 32492871698, 62838521438, 117376191548, 212507783895, 374035542815, 641676187295, 1075368660795, 1763954792145

Offset: 1

Johannes W. Meijer and Nico Baken (n.h.g.baken(AT)tudelft.nl), Jun 18 2009

T. D. Noe, Table of n, a(n) for n = 1..1000

Equals fifth right hand column of A161739 (RSEG2 triangle).

Other right hand columns are A000292, A107963, A161740.

a(n) = (n+5)*(n+4)*(n+3)*(n+2)*(n+1)*(n)*(175*n^6+1365*n^5+2671*n^4-2805*n^3-14735*n^2-14445*n-2466)/5443200.

The denominator of a(n) (5443200) equals the product of the first four terms of A036283 (6*60*126*120). - Johannes W. Meijer, Oct 22 2009

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

Original entry on oeis.org

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

A163929 Denominators of the higher order exponential integral constants alpha(2,n).

Original entry on oeis.org

1, 1, 16, 1296, 20736, 12960000, 4320000, 10372320000, 165957120000, 40327580160000, 40327580160000, 590436101122560000, 590436101122560000, 16863445484161436160000, 2409063640594490880000, 2409063640594490880000

Offset: 1

Johannes W. Meijer & Nico Baken, Aug 13 2009, Aug 17 2009

See A163927 for information about the constants alpha(k,n).

			alpha(k=2, n=1) = 0, alpha(k=2, 2) = 1, alpha(k=2, 3) = 21/16, and alpha(k=2, 4) = 1897/1296.

Cf. A163928 (numerators).

alpha(k,n) = (1/k)*Sum_{i=0..k-1} Sum_{p=0..n-1} p^(-2*(k-i))*alpha(i, n) with alpha(0,n) = 1. For this sequence, k = 2 and n >= 1.

A161739 The RSEG2 triangle.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 4, 1, 0, 0, 13, 10, 1, 0, -4, 30, 73, 20, 1, 0, 0, -14, 425, 273, 35, 1, 0, 120, -504, 1561, 3008, 798, 56, 1, 0, 0, 736, -2856, 25809, 14572, 1974, 84, 1, 0, -12096, 44640, -73520, 125580, 218769, 55060, 4326, 120, 1

Offset: 0

Johannes W. Meijer & Nico Baken (n.h.g.baken(AT)tudelft.nl), Jun 18 2009

The EG2[2*m,n] matrix coefficients were introduced in A008955. We discovered that EG2[2m,n] = Sum_{k = 1..n} (-1)^(k+n)*t1(n-1,k-1)*2*eta(2*m-2*n+2*k)/((n-1)!)^2 with t1(n,m) the central factorial numbers A008955 and eta(m) = (1-2^(1-m))*zeta(m) with eta(m) the Dirichlet eta function and zeta(m) the Riemann zeta function.
A different way to define these matrix coefficients is EG2[2*m,n] = (1/m)*Sum_{k = 0..m-1} ZETA(2*m-2*k, n-1)*EG2[2*k, n] with ZETA(2*m, n-1) = zeta(2*m) - Sum_{k = 1..n-1} (k)^(-2*m) and EG2[0, n] = 1, for m = 0, 1, 2, ..., and n = 1, 2, 3, ... .
We define the row sums of the EG2 matrix rs(2*m,p) = Sum_{n >= 1} (n^p)*EG2(2*m,n) for p = -2, -1, 0, 1, ... and m >= p+2. We discovered that rs(2*m,p=-2) = 2*eta(2*m+2) = (1 - 2^(1-(2*m+2)))*zeta(2*m+2). This formula is quite unlike the other rs(2*m,p) formulas, see the examples.
The series expansions of the row generators RGEG2(z,2*m) about z = 0 lead to the EG2[2*m,n] coefficients while the series expansions about z = 1 lead to the ZG1[2*m-1,n] coefficients, see the formulas.
The first Maple program gives the triangle coefficients. Adding the second program to the first one gives information about the row sums rs(2*m,p).
The a(n) formulas of the right hand columns are related to sequence A036283, see also A161740 and A161741.

			The first few expressions for the ZG1[2*m-1,p+1] coefficients are:
  ZG1[2*m-1, 1] = (zeta(2*m-1))/(1/2)
  ZG1[2*m-1, 2] = (zeta(2*m-3) - zeta(2*m-1))/1
  ZG1[2*m-1, 3] = (zeta(2*m-5) - 5*zeta(2*m-3) + 4*zeta(2*m-1))/6
  ZG1[2*m-1, 4] = (zeta(2*m-7) - 14*zeta(2*m-5) + 49*zeta(2*m-3) - 36*zeta(2*m-1))/72
The first few rs(2*m,p) are (m >= p+2)
  rs(2*m, p=0) = ZG1[2*m-1,1]
  rs(2*m, p=1) = ZG1[2*m-1,1] + ZG1[2*m-1,2]
  rs(2*m, p=2) = ZG1[2*m-1,1] + 3*ZG1[2*m-1,2] + 2*ZG1[2*m-1,3]
  rs(2*m, p=3) = ZG1[2*m-1,1] + 7*ZG1[2*m-1,2] + 12*ZG1[2*m-1,3] + 6*ZG1[2*m-1,4]
The first few rs(2*m,p) are (m >= p+2)
  rs(2*m, p=-1) = zeta(2*m+1)/(1/2)
  rs(2*m, p=0) = zeta(2*m-1)/(1/2)
  rs(2*m, p=1) = (zeta(2*m-1) + zeta(2*m-3))/1
  rs(2*m, p=2) = (zeta(2*m-1) + 4*zeta(2*m-3) + zeta(2*m-5))/3
  rs(2*m, p=3) = (0*zeta(2*m-1) + 13*zeta(2*m-3) + 10*zeta(2*m-5) + zeta(2*m-7))/12
The first few rows of the RSEG2 triangle are:
  [1]
  [0, 1]
  [0, 1, 1]
  [0, 1, 4, 1]
  [0, 0, 13, 10, 1]
  [0, -4, 30, 73, 20, 1]

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, Chapter 23, pp. 811-812.
J. W. Meijer and N.H.G. Baken, The Exponential Integral Distribution, Statistics and Probability Letters, Volume 5, No.3, April 1987. pp 209-211.

A000007, A129825, A161742 and A161743 are the first four left hand columns.
A000012, A000292, A107963, A161740 and A161741 are the first five right hand columns.
A010790 equals 2*r(n) and A054977 equals denom(r(n)).
A001710 equals numer(q(n)) and A141044 equals denom(q(n)).
A000142 equals the row sums.
A008955 is a central factorial number triangle.
A028246 is Worpitzky's triangle.

nmax:=10; for n from 0 to nmax do A008955(n, 0) := 1 end do: for n from 0 to nmax do A008955(n, n) := (n!)^2 end do: for n from 1 to nmax do for m from 1 to n-1 do A008955(n, m) := A008955(n-1, m-1)*n^2 + A008955(n-1, m) end do: end do: for n from 1 to nmax do A028246(n, 1) := 1 od: for n from 1 to nmax do A028246(n, n) := (n-1)! od: for n from 3 to nmax do for m from 2 to n-1 do A028246(n, m) := m*A028246(n-1, m) + (m-1)*A028246(n-1, m-1) od: od: for i from 0 to nmax-2 do s(i) := ((i+1)!/2)*sum(A028246(i+1, k1+1)*(sum((-1)^(j)*A008955(k1, j)*2*x^(2*nmax-(2*k1+1-2*j)), j=0..k1)/ (k1!*(k1+1)!)), k1=0..i) od: a(0,0) := 1: for n from 1 to nmax-1 do for m from 0 to n do a(n,m) := coeff(s(n-1), x, 2*nmax-1-2*m+2) od: od: seq(seq(a(n, m), m=0..n), n=0..nmax-1); for n from 0 to nmax-1 do seq(a(n, m), m=0..n) od;
m:=7: row := 2*m; rs(2*m, -2) := 2*eta(2*m+2); for p from -1 to m-2 do q(p+1) := (p+1)!/2 od: for p from -1 to m-2 do rs(2*m, p) := sum(a(p+1, k)*zeta(2*m+1-2*k), k=0..p+1)/q(p+1) od;

RGEG2(2*m,z) = Sum_{n >= 1} EG2[2*m,n]*z^(n-1) = Integral_{y = 0..oo}((2*y)^(2*m)/(2*m)!)* cosh(y)/(cosh(y)^2 - z)^(3/2) for m >= 0.
EG2[2*m,n] = Sum_{k = 1..n} (-1)^(k+n)* A008955(n-1, k-1)*2*eta(2*m-2*n+2*k)/((n-1)!)^2.
ZG1[2*m-1,p+1] = Sum_{j = 0..p} (-1)^j*A008955(p, j)*zeta(2*m-(2*p+1-2*j))/ r(p) with r(p)= p!*(p+1)!/2 and p >= 0.
rs(2*m,p) = Sum_{k = 0..p} A028246(p+1,k+1)*ZG1[2*m-1,k+1] and p >= 0; p <= m-2.
rs(2*m,p) = Sum_{k = 0..p+1} A161739(p+1,k)*zeta(2*m+1-2*k)/q(p+1) with q(p+1) = (p+1)!/2 and p >= -1; p <= m-2.
From Peter Bala, Mar 19 2022: (Start)
It appears that the k-th row polynomial (with indexing starting at k = 1) is given by R(k,n^2) = (k-1)!*Sum_{i = 0..n} (-1)^(n-i)*(i^k)* binomial(n,i)*binomial(n+i,i)/(n+i) for n >= 1.
For example, for k = 6, Maple's SumTools:-Summation procedure gives 5!*Sum_{i = 0..n} (-1)^(n-i)*(i^6)*binomial(n,i)*binomial(n+i,i)/(n+i) = -4*n^2 + 30*n^4 + 73*n^6 + 20*n^8 + n^10 = R(6,n^2). (End)

Minor error corrected and edited by Johannes W. Meijer, Sep 22 2012

A163928 Numerators of the higher order exponential integral constants alpha(2,n).

Original entry on oeis.org

0, 1, 21, 1897, 32197, 20881861, 7139587, 17462165587, 283355376967, 69621962857381, 70246946681461, 1036088178214798501, 1042504974775473001, 29931734181763981573561, 4295332813075795410223, 4312254507400142830831

Offset: 1

Johannes W. Meijer & Nico Baken, Aug 13 2009, Aug 17 2009

See A163927 for information about the alpha(k,n) constants.

Apart from a difference of offset, alpha(2,n) appears to be the multiple harmonic (star) sum Sum_{j = 1..n} 1/j^2 Sum_{k = 1..j} 1/k^2, which has the initial values [1, 21/16, 1897/1296, 32197/20736, 20881861/12960000, 7139587/4320000, ...]. - Peter Bala, Jan 31 2019

			alpha(k=2,n=1) = 0, alpha(k=2,2) = 1, alpha(k=2,3) = 21/16, alpha(k=2,4) = 1897/1296.

Cf. A163929 (denominators).
Cf. A163927 (alpha(k,n)) and A090998 (gamma(k,n)).
Cf. A007406, A007407.

nmax:=17; rowk:=2; kmax:=nmax: k:=0: for n from 1 to nmax do alpha(k,n):=1 od: for k from 1 to kmax do for n from 1 to nmax do alpha(k,n) := (1/k)*sum(sum(p^(-2*(k-i)),p=0..n-1)*alpha(i, n),i=0..k-1) od; od: seq(alpha(rowk, n),n=1..nmax);

alpha(k,n) = (1/k)*Sum_{i=0..k-1} (Sum_{p=0..n-1} p^(-2*(k-i))*alpha(i, n) with alpha(0,n) = 1, with k = 2 and n >= 1. alpha(1,n) = A007406(n-1)/A007407(n-1) for n >= 2.

A163931 Decimal expansion of the higher-order exponential integral E(x, m=2, n=1) at x=1.

Original entry on oeis.org

0, 9, 7, 8, 4, 3, 1, 9, 7, 2, 1, 6, 6, 7, 0, 1, 7, 9, 3, 2, 5, 5, 3, 7, 7, 8, 9, 0, 4, 5, 2, 8, 0, 0, 8, 2, 7, 6, 9, 5, 8, 2, 2, 6, 9, 5, 3, 0, 2, 6, 5, 7, 6, 5, 5, 7, 4, 4, 2, 1, 2, 4, 2, 4, 5, 4, 4, 7, 1, 3, 7, 6, 2, 6, 1, 4, 0, 9, 0, 4, 8, 8, 7, 3, 6, 9, 6, 0, 4, 8, 9, 1, 8, 5, 5, 5, 0, 8, 9, 4, 5, 4, 6, 7, 0

Offset: 0

Johannes W. Meijer and Nico Baken, Aug 13 2009, Aug 17 2009

We define the higher-order exponential integrals by E(x,m,n) = x^(n-1)*Integral_{t=x..infinity} E(t,m-1,n)/t^n for m >= 1 and n >= 1 with E(x,m=0,n) = exp(-x), see Meijer and Baken.
The properties of the E(x,m,n) are analogous to those of the well-known exponential integrals E(x,m=1,n), see Abramowitz and Stegun and the formulas.
The series expansions of the higher-order exponential integrals are dominated by the constants alpha(k,n), see A163927, and gamma(k,n) = G(k,n), see A090998.
For information about the asymptotic expansion of the E(x,m,n) see A163932.
Values of E(x,m,n) can be evaluated with the Maple program.

			E(1,2,1) = 0.09784319721667017932553778904528008276958226953026576557442124245....

G. C. Greubel, Table of n, a(n) for n = 0..5000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, Chapter 5, pp. 227-251.
J. W. Meijer and N. H. G. Baken, The Exponential Integral Distribution, Statistics and Probability Letters, Volume 5, No.3, April 1987. pp 209-211.
M. S. Milgram, The generalized integro-exponential function, Math. of Computation, Vol. 44, pp. 443-458, 1985.
Eric Weisstein's World of Mathematics, The Exponential Integral.

Cf. A163927 (alpha(k,n)), A090998 (gamma(k,n) = G(k,n)), A163932.
Cf. A068985 (E(x=1,m=0,n) = exp(-1)) and A099285 (E(x=1,m=1,n=1)).
Cf. A001563 (n*n!), A002775 (n^2*n!), A091363 (n^3*n!) and A091364 (n^4*n!).

E:= proc(x,m,n) local nmax, kmax, EI, k1, k2, n1, n2; option remember: nmax:=20; kmax:=20; k1:=0: for n1 from 0 to nmax do alpha(k1,n1):=1 od: for k1 from 1 to kmax do for n1 from 1 to nmax do alpha(k1,n1) := (1/k1)*sum(sum(p^(-2*(k1-i1)),p=0..n1-1)*alpha(i1, n1),i1=0..k1-1) od; od: for n2 from 0 to kmax do G(0,n2):=1 od: for n2 from 1 to nmax do for k2 from 1 to kmax do G(k2,n2):=(1/k2)*(((gamma-sum(p^(-1),p=1..n2-1))*G(k2-1,n2)+ sum((Zeta(k2-i2)-sum(p^(-(k2-i2)), p=1..n2-1))*G(i2,n2),i2=0..k2-2))) od; od: EI:= evalf((-1)^m*((-x)^(n-1)/(n-1)!*sum(alpha(kz,n)*(G(m-2*kz,n)+sum(G(m-2*kz-i,n)*ln(x)^i/i!,i=1..m-2*kz)), kz=0..floor(m/2)) + sum((-x)^kx/((kx-n+1)^m*kx!),kx=0..n-2) + sum((-x)^ky/((ky-n+1)^m*ky!),ky=n..infinity))); return(EI): end:

Join[{0}, RealDigits[ N[ EulerGamma^2/2 + Pi^2/12 - HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, -1], 104]][[1]]] (* Jean-François Alcover, Nov 07 2012, from 1st formula *)

t=1; Euler^2/2 + Pi^2/12 + sumalt(k=1, t*=k; (-1)^k/(k^2*t)) \\ Charles R Greathouse IV, Nov 07 2016

E(x=1,m=2,n=1) = gamma^2/2 + Pi^2/12 + Sum_{k>=1} ((-1)^k/(k^2*k!)).
E(x=0,n,m) = (1/(n-1))^m for n >= 2.
Integral_{t=0..x} E(t,m,n) = 1/n^m - E(x,n,n+1).
dE(x,m,n+1)/dx = - E(x,m,n).
E(x,m,n+1) = (1/n)*(E(x,m-1,n+1) - x*E(x,m,n)).
E(x,m,n) = (-1)^m * ((-x)^(n-1)/(n-1)!) * Sum_{kz=0..floor(m/2)}(alpha (kz, n)*G(m-2*kz, n)) + (-1) ^m * ((-x)^(n-1)/(n-1)!) * Sum_{kz=0..floor(m/2)}(Sum_{i=1..m-2*kz}(alpha (kz, n) *G(m-2*kz-i, n)*log(x)^i/i!)) + (-1)^m * Sum_{ kx=0..n-2}((-x)^kx/((kx-n+1)^m*kx!) + (-1)^m * Sum_{ky>=n}((-x)^ky /(( ky-n+1)^m*ky!)).

A163927 Numerators of the higher order exponential integral constants alpha(k,4).

Original entry on oeis.org

1, 49, 1897, 69553, 2515513, 90663937, 3264855049, 117543378001, 4231639039705, 152339702576545, 5484235568128681, 197432536935184369, 7107571838026381177, 255872590744254526273, 9211413307971174616393

Offset: 0

Johannes W. Meijer and Nico Baken, Aug 13 2009, Aug 17 2009

The higher order exponential integrals, see A163931, are defined by E(x,m,n) = x^(n-1)*Integral_{t>=x} E(t,m-1,n)/t^n for m >= 1 and n >= 1, with E(x,m=0,n) = exp(-x).
The series expansions of the higher order exponential integrals are dominated by the alpha(k,n) and the gamma(k,n) constants, see A090998.
The first Maple program uses the alpha(k,n) formula and the second the GF(z,n) to generate the alpha(k,n) coefficients in each column.
Appears to equal the numerator of the multiple harmonic (star) sum Sum_{1 <= k_1 <= ... <= k_n <= 3} 1/(k_1^2*...*k_n^2). If true, then a(n) = numerator( 3/2 - 3/(5*4^n) + 1/(10*9^n) ). - Peter Bala, Jan 31 2019

			a(k=0,n=4) = 1, a(k=1,4) = 49/36, a(k=2,4) = 1897/1296, a(k=3,4) = 69553/46656.

J. W. Meijer and N. H. G. Baken, The Exponential Integral Distribution, Statistics and Probability Letters, Volume 5, No. 3, April 1987, pp. 209-211.

Cf. A163931 (E(x,m,n)), A090998 (gamma(k,n)).
Cf. A163928 y A163929.
a(k,1) = A000007(k)
a(k,2) = A000012(k) = 1^k.
a(k,3) = A002450(k+1)/A000302(k) with A000302(k) = 4^k.
a(k,4) = A163927(k)/A009980(k) with A009980(k) = 36^k.
The GF(z,n) lead to A008955.
The denominators of a(1,n), n >= 2, lead to A007407.

coln := 4; nmax := 15; kmax := nmax: k:=0: for n from 1 to nmax do alpha(k, n) := 1 od: for k from 1 to kmax do for n from 1 to nmax do alpha(k, n) := (1/k)*sum(sum(p^(-2*(k-i)), p=0..n-1)*alpha(i, n), i=0..k-1) od; od: seq(alpha(k, coln), k=0..nmax-1);
# End program 1
coln:=4; nmax1 := 16; for n from 0 to nmax1 do A008955(n, 0):=1 end do: for n from 0 to nmax1 do A008955(n, n) := (n!)^2 end do: for n from 1 to nmax1 do for m from 1 to n-1 do A008955(n, m) := A008955(n-1, m-1)*n^2 + A008955(n-1, m) end do: end do: m:=coln-1: f(m):=0: for n from 0 to m do f(m) := f(m) + (-1)^(n + m)*A008955(m, n)*z^(2*m-2*n) od: GF(z,coln) := m!^2/f(m): GF(z,coln):=series(GF(z,coln), z, nmax1);
# End program 2

alpha(k,n) = (1/k) * Sum_{i=0..k-1} (Sum_{p=0..n-1}(p^(2*i-2*k))*alpha(i, n)) with alpha(0,n) = 1, k >= 0 and n >= 1.
alpha(k,n) = alpha(k,n+1) -alpha(k-1,n+1)/n^2.
GF(z,n) = product((1-(z/k)^2)^(-1), k = 1..n-1) = (Pi*z/sin(Pi*z))/(Beta(n+z,n-z)/Beta(n,n)).

A161740 Fourth right hand column of the RSEG2 triangle A161739.

Original entry on oeis.org

0, 30, 425, 3008, 14572, 55060, 174130, 481360, 1197196, 2733874, 5817955, 11668800, 22250280, 40616264, 71373956, 121292960, 200092040, 321439910, 504211037, 774042368, 1165242100, 1723107100, 2506711350, 3592233840

Offset: 1

Johannes W. Meijer & Nico Baken (n.h.g.baken(AT)tudelft.nl), Jun 18 2009

Equals fourth right hand column of A161739 (RSEG2 triangle).

Other right hand columns are A000292, A107963, A161741.

a(n) = (n+4)*(n+3)*(n+2)*(n+1)*(n)*(n-1)*(35*n^3+189*n^2+325*n+204)/45360

The denominator of a(n) (45360) equals the product of the first three terms of A036283 (6*60*126). - Johannes W. Meijer, Oct 22 2009

A161742 Third left hand column of the RSEG2 triangle A161739.

Original entry on oeis.org

1, 4, 13, 30, -14, -504, 736, 44640, -104544, -10644480, 33246720, 5425056000, -20843695872, -5185511654400, 23457840537600, 8506857655296000, -44092609863966720, -22430879475779174400, 130748316971139072000

Offset: 2

Johannes W. Meijer & Nico Baken (n.h.g.baken(AT)tudelft.nl), Jun 18 2009

Equals third left hand column of A161739 (RSEG2 triangle).
Other left hand columns are A129825 and A161743.
A008955 is a central factorial number triangle.
A028246 is Worpitzky's triangle.
A001710 (n!/2!), A001715 (n!/3!), A001720 (n!/4!), A001725 (n!/5!), A001730 (n!/6!), A049388 (n!/7!), A049389 (n!/8!), A049398 (n!/9!), A051431 (n!/10!) appear in Maple program.

nmax:=21; for n from 0 to nmax do A008955(n,0):=1 end do: for n from 0 to nmax do A008955(n,n):=(n!)^2 end do: for n from 1 to nmax do for m from 1 to n-1 do A008955(n,m):= A008955(n-1,m-1)*n^2+A008955(n-1,m) end do: end do: for n from 1 to nmax do A028246(n,1):=1 od: for n from 1 to nmax do A028246(n,n):=(n-1)! od: for n from 3 to nmax do for m from 2 to n-1 do A028246(n,m):=m*A028246(n-1,m)+(m-1)*A028246(n-1,m-1) od: od: for n from 2 to nmax do a(n):=sum(((-1)^k/((k+1)!*(k+2)!)) *(n!)*A028246(n,k+2)* A008955(k+1,k),k=0..n-2) od: seq(a(n),n=2..nmax);

a(n) = sum(((-1)^k/((k+1)!*(k+2)!))*(n!)*A028246(n, k+2)*A008955(k+1, k), k=0..n-2)

cp's OEIS Frontend

User: Nico Baken

A161743 Fourth left hand column of the RSEG2 triangle A161739.

Views

Author

Keywords

Crossrefs

Programs

Maple

Formula

A161741 Fifth right hand column of the RSEG2 triangle A161739.

Views

Author

Keywords

Links

Crossrefs

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A163929 Denominators of the higher order exponential integral constants alpha(2,n).

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A161739 The RSEG2 triangle.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula

Extensions

A163928 Numerators of the higher order exponential integral constants alpha(2,n).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Formula

A163931 Decimal expansion of the higher-order exponential integral E(x, m=2, n=1) at x=1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A163927 Numerators of the higher order exponential integral constants alpha(k,4).

Views

Author

Keywords

Comments

Examples