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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A063395 T(2n,n) with T(n,m) as in A063394.

Original entry on oeis.org

1, 3, 19, 131, 979, 7683, 62099, 511619, 4271699, 36018179, 305998099, 2615234691, 22459983059, 193665818115, 1675580699539, 14538892408451, 126467748738899, 1102484411211779, 9629327766561299, 84247346901823619, 738200425192338899, 6477139329614712323
Offset: 0

Views

Author

Floor van Lamoen, Jul 16 2001

Keywords

Crossrefs

Equals 4*A084771(n-1) - 1, n>0.

Programs

  • PARI
    m=matrix(50,50):for(i=1,50,m[1,i]=1:m[i,1]=1):for(i=2,50, for(k=1,i,x=i-k+1: if(m[x,k]==0,m[x,k]=sum(n=2,k-1,m[x,n])+sum(n=2,x-1,m[n,k])+k+x-1))):for(n=1,24,print1(m[n,n]",")) /* Ralf Stephan */

Formula

G.f.: 4x/sqrt(1-10x+9x^2) + (1-2x)/(1-x). - Ralf Stephan, Mar 23 2004
Conjecture: (-n+1)*a(n) +(11*n-17)*a(n-1) +(-19*n+43)*a(n-2) +9*(n-3)*a(n-3)=0. - R. J. Mathar, Jun 10 2013

Extensions

More terms from Ralf Stephan, Mar 23 2004

A363571 Expansion of (1 / sqrt(1 - 10*x + 9*x^2) - 1 / (1 - x)) / 4.

Original entry on oeis.org

0, 1, 8, 61, 480, 3881, 31976, 266981, 2251136, 19124881, 163452168, 1403748941, 12104113632, 104723793721, 908680775528, 7904234296181, 68905275700736, 601832985410081, 5265459181363976, 46137526574521181, 404821208100919520, 3556361565584509001
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 17 2023

Keywords

Crossrefs

Cf. A030662, A047665, A084771, A098665, A363570.

Programs

  • Mathematica
    nmax = 21; CoefficientList[Series[(1/Sqrt[1 - 10 x + 9 x^2] - 1/(1 - x))/4, {x, 0, nmax}], x]
nmax = 21; CoefficientList[Series[Exp[x] (Exp[4 x] BesselI[0, 4 x] - 1)/4, {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Binomial[n, k]^2 4^(k - 1), {k, 1, n}], {n, 0, 21}]
Table[(3^n LegendreP[n, 5/3] - 1)/4, {n, 0, 21}]

Formula

E.g.f.: exp(x) * (exp(4*x) * BesselI(0,4*x) - 1) / 4.
a(n) = Sum_{k=1..n} binomial(n,k)^2 * 4^(k-1).
a(n) = (3^n * LegendreP(n,5/3) - 1) / 4.
a(n) = (A084771(n) - 1) / 4.

A383132 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n*k,k) * n^k.

Original entry on oeis.org

1, 2, 33, 2701, 524993, 181752001, 97735073905, 75179269556672, 78240951854025217, 105806762566689176353, 180297512864534759056001, 377878889913778527874694227, 955217573424445946022789385537, 2865620569274978738097814056365899, 10064763360358683666070320479027168465
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 17 2025

Keywords

Crossrefs

Cf. A084771, A187021, A331656, A340971, A383120, A383133.

Programs

  • Mathematica
    Unprotect[Power]; 0^0 = 1; Table[Sum[Binomial[n, k] Binomial[n k, k] n^k, {k, 0, n}], {n, 0, 14}]

Formula

a(n) = [x^n] ((1 + n*x)^n + x)^n.
a(n) ~ exp(n - 1/2) * n^(2*n - 1/2) / sqrt(2*Pi). - Vaclav Kotesovec, Apr 19 2025
Previous Showing 21-23 of 23 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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