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.

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A051565 Third unsigned column of triangle A051523.

Original entry on oeis.org

0, 0, 1, 33, 791, 17100, 358024, 7491484, 159168428, 3463513704, 77559615576, 1792139785920, 42789106278720, 1056302350122240, 26964471256888320, 711643650545422080, 19410244660543737600, 546854985563699289600
Offset: 0

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Author

Wolfdieter Lang

Keywords

Comments

From Johannes W. Meijer, Oct 20 2009: (Start)
The asymptotic expansion of the higher order exponential integral E(x,m=3,n=10) ~ exp(-x)/x^3*(1 - 33/x + 791/x^2 - 17100/x^3 + 358024/x^4 + ...) leads to the sequence given above. See A163931 and A163932 for more information.
(End)

References

  • Mitrinovic, D. S. and Mitrinovic, R. S. see reference given for triangle A051523.

Crossrefs

Cf. A049398 (m=0), A051564 (m=1) unsigned columns.

Formula

a(n) = A051523(n, 2)*(-1)^n; e.g.f.: ((log(1-x))^2)/(2*(1-x)^10).
If we define f(n,i,a)=sum(binomial(n,k)*stirling1(n-k,i)*product(-a-j,j=0..k-1),k=0..n-i), then a(n) = |f(n,2,10)|, for n>=1. - Milan Janjic, Dec 21 2008

A163930 Duplicate of A090998.

Original entry on oeis.org

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

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Author

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

Keywords

Comments

The higher order exponential integrals, see A163931, are defined by E(x,m,n) = x^(n-1)*int(E(t,m-1,n)/t^n, t=x..infinity) 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 gamma(k,n) and the alpha(k,n) constants, see A163927.
The values of the gamma(k,n) = G(k,n) coefficients can be determined with the Maple program.

Examples

			G(2,1) = 0.9890559953279725553953956515...

Links

Crossrefs

Cf. A163931 (E(x,m,n)), A163927 (alpha(k,n)).
G(1,1) equals A001620 (gamma).
(gamma - G(1,n)) equals A001008(n-1)/A002805(n-1) for n>=2.
The structure of the G(k,n=1) formulas lead (replace gamma by G and Zeta by Z) to A036039.

Programs

  • Maple
    ncol:=1; nmax:=5; kmax:=nmax; for n from 1 to nmax do G(0,n):=1 od: for n from 1 to nmax do for k from 1 to kmax do G(k,n):= expand((1/k)*((gamma-sum(p^(-1),p=1..n-1))* G(k-1,n)+sum((Zeta(k-i)-sum(p^(-(k-i)),p=1..n-1))*G(i,n),i=0..k-2))) od; od: for k from 0 to kmax do G(k,ncol):=G(k,ncol) od;
  • Mathematica
    RealDigits[ N[ EulerGamma^2/2 + Pi^2/12, 105]][[1]] (* Jean-François Alcover, Nov 07 2012, from 1st formula *)

Formula

G(2,1) = gamma(2,1) = gamma^2/2+Pi^2/12.
G(k,n) = (1/k)*(gamma*G(k-1,n)) - (1/k)*Sum_{p=1..n-1}(p^(-1))* G(k-1,n) + (1/k)* Sum_{i=0..k-2}(Zeta(k-i) * G(i,n)) - (1/k)*Sum_{i=0..k-2}(Sum_{p=1..n-1}(p^(i-k)) * G(i,n)) with G(0,n) = 1 for k>=0 and n>=1.
G(k,n+1) = G(k,n) -G(k-1,n)/n.
GF(z,n) = GAMMA(n-z)/GAMMA(n).

A273878 Numerator of (2*(n+1)!/(n+2)).

Original entry on oeis.org

1, 4, 3, 48, 40, 1440, 1260, 8960, 72576, 7257600, 6652800, 958003200, 889574400, 11623772160, 163459296000, 41845579776000, 39520825344000, 12804747411456000, 12164510040883200, 231704953159680000, 4644631106519040000
Offset: 0

Views

Author

Johannes W. Meijer, Jun 08 2016

Keywords

Comments

The moments, i.e. E(X^n) = int(x^n * p(x), x = 0..infinity) for n > 0, of the probability density function p(x) = 2*x*E(x, 1, 1), see A163931, lead to this sequence.

Examples

			The first few moments of p(x) are: 1, 4/3, 3, 48/5, 40, 1440/7, … .

Links

Crossrefs

Cf. A090585 (denominators), A090586, A085250, A110468, A128059, A128060, A163931.

Programs

  • Maple
    a := proc(n): numer(2*(n+1)!/(n+2)) end: seq(a(n), n=0..20);
  • PARI
    a(n) = numerator(2*(n+1)!/(n+2)) \\ Felix Fröhlich, Jun 09 2016

Formula

a(n) = numer(2*(n+1)!/(n+2))
a(n) = (n+1) * A090586(n+1)
a(2*n) = A110468(n) and a(2*n+1) = (2*n)!*A085250(n+1)/A128060(n+2).

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

Views

Author

Igor Victorovich Statsenko, Sep 30 2024

Keywords

Comments

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.

Examples

			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;

Links

Crossrefs

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.

Programs

  • Maple
    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);
Previous Showing 71-74 of 74 results.

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