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.

Showing 1-3 of 3 results.

A051893 a(n) = Sum_{i=1..n-1} i^2*a(i), a(1) = 1.

Original entry on oeis.org

1, 1, 5, 50, 850, 22100, 817700, 40885000, 2657525000, 217917050000, 22009622050000, 2685173890100000, 389350214064500000, 66189536390965000000, 13039338669020105000000, 2946890539198543730000000, 757350868574025738610000000, 219631751886467464196900000000
Offset: 1

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Author

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 17 1999

Keywords

Crossrefs

Cf. A001710, A101686, A256019, A256020.

Programs

  • Maple
    a := n -> `if`(n=1,1,(sinh(Pi)*GAMMA(n-I)*GAMMA(n+I))/(2*Pi)):
seq(simplify(a(n)), n=1..18); # Peter Luschny, Oct 19 2016
  • Mathematica
    a[n_] := Pochhammer[2-I, n-2]*Pochhammer[2+I, n-2]; a[1] = 1; Table[a[n], {n, 1, 15}] (* Jean-François Alcover, Dec 21 2012, after Vladeta Jovovic *)
Join[{1},FoldList[Times,1,Range[2,20]^2+1]] (* Harvey P. Dale, Jul 04 2013 *)
Clear[a]; a[1]=1; a[n_]:=a[n]=Sum[i^2*a[i],{i,1,n-1}]; Table[a[n],{n,1,20}] (* Vaclav Kotesovec, Mar 13 2015 *)

Formula

a(n) = Product_{i=2..n-1} (i^2+1), for n>2. - Vladeta Jovovic, Nov 26 2002
From Vaclav Kotesovec, Mar 13 2015: (Start)
For n > 1, a(n) = A101686(n-1)/2.
a(n) ~ (n-1)!^2 * sinh(Pi)/(2*Pi).
(End)
a(n) = (A003703(n)^2 + A009454(n)^2 + A000007(n-1))/2. - Vladimir Reshetnikov, Oct 15 2016
a(n) = sinh(Pi)*Gamma(n-I)*Gamma(n+I)/(2*Pi) for n>1. - Peter Luschny, Oct 19 2016

Extensions

More terms from Harvey P. Dale, Jul 04 2013

A256019 a(n) = Sum_{i=1..n-1} (i^3 * a(i)), a(1)=1.

Original entry on oeis.org

1, 1, 9, 252, 16380, 2063880, 447861960, 154064514240, 79035095805120, 57695619937737600, 57753315557675337600, 76927416322823549683200, 133007502822161917402252800, 292350491203111894450151654400, 802502098352542150265666291328000
Offset: 1

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Author

Vaclav Kotesovec, Mar 13 2015

Keywords

Comments

a(n) = A158621(n-1) for n > 2. - Georg Fischer, Oct 23 2018

Crossrefs

Cf. A001710, A051893, A158621, A256020.

Programs

  • Mathematica
    Clear[a]; a[1]=1; a[n_]:= a[n] = Sum[i^3*a[i],{i,1,n-1}]; Table[a[n],{n,1,20}]
Flatten[{1, Table[FullSimplify[Cosh[Sqrt[3]*Pi/2] * Gamma[n+1] * Gamma[n-1/2 - I*Sqrt[3]/2] * Gamma[n-1/2 + I*Sqrt[3]/2] / (2*Pi)],{n,2,20}]}]

Formula

Product_{i=2..n-1} (i^3 + 1), for n>2.
a(n) ~ cosh(sqrt(3)*Pi/2) / (2*Pi) * ((n-1)!)^3.
a(n) = A255433(n-1)/2.

A376097 a(0) = 1; a(n) = Sum_{k=0..n-1} (k+1)^4 * a(k) * a(n-k-1).

Original entry on oeis.org

1, 1, 17, 1410, 364019, 228282823, 296324235500, 712075198644414, 2918094100584013255, 19151474626728425949663, 191553141880332262049655201, 2804913258838830873001491036584, 58168297154586087400230338311689652, 1661461159115675581245556180230933084340
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 10 2024

Keywords

Crossrefs

Cf. A000699, A015085, A088716, A256020, A376095, A376096.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = Sum[(k + 1)^4 a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 13}]
nmax = 13; A[] = 0; Do[A[x] = 1 + x A[x]^2 + 15 x^2 A[x] A'[x] + 25 x^3 A[x] A''[x] + 10 x^4 A[x] A'''[x] + x^5 A[x] A''''[x] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

Formula

G.f. A(x) satisfies: A(x) = 1 + x * A(x)^2 + 15 * x^2 * A(x) * A'(x) + 25 * x^3 * A(x) * A''(x) + 10 * x^4 * A(x) * A'''(x) + x^5 * A(x) * A''''(x).
Showing 1-3 of 3 results.

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

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