A256019 -id:A256019 - cp's OEIS frontend

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

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

Offset: 1

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

Cf. A001710, A101686, A256019, A256020.

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

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 *)

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

More terms from Harvey P. Dale, Jul 04 2013

A256020 a(n) = Sum_{i=1..n-1} (i^4 * a(i)), a(1)=1.

Original entry on oeis.org

1, 1, 17, 1394, 358258, 224269508, 290877551876, 698687879606152, 2862524242746404744, 18783884080901907930128, 187857624693099981209210128, 2750611340756369924865254694176, 57039427373264843131930786593127712, 1629160124635190449534207126672913710144

Offset: 1

Vaclav Kotesovec, Mar 13 2015

nonn

Cf. A001710, A051893, A255434, A256019.

Clear[a]; a[1]=1; a[n_]:= a[n] = Sum[i^4*a[i],{i,1,n-1}]; Table[a[n],{n,1,15}]
Flatten[{1,1, Table[Product[(i^4 + 1), {i,2,n-1}],{n,3,15}]}]
Join[{1},FoldList[Times,Range[15]^4+1]/2] (* Harvey P. Dale, Jul 29 2018 *)

a(n) = Product_{i=2..n-1} (i^4 + 1), for n>2.
a(n) ~ (cosh(Pi/sqrt(2))^2 * sin(Pi/sqrt(2))^2 + cos(Pi/sqrt(2))^2 * sinh(Pi/sqrt(2))^2) / (2*Pi^2) * ((n-1)!)^4.
a(n) = A255434(n-1)/2.

A376096 a(0) = 1; a(n) = Sum_{k=0..n-1} (k+1)^3 * a(k) * a(n-k-1).

Original entry on oeis.org

1, 1, 9, 260, 17215, 2189997, 477731884, 164858203944, 84745577983095, 61951785517193675, 62077057930391945969, 82749694746019635920952, 143157935882304543684640676, 314805573970543375502985796300, 864458294787075036217714712292600, 2919280453922974335841433174057739408

Offset: 0

Ilya Gutkovskiy, Sep 10 2024

nonn

Cf. A000699, A015084, A088716, A256019, A376095, A376097.

a[0] = 1; a[n_] := a[n] = Sum[(k + 1)^3 a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 15}]
nmax = 15; A[] = 0; Do[A[x] = 1 + x A[x]^2 + 7 x^2 A[x] A'[x] + 6 x^3 A[x] A''[x] + x^4 A[x] A'''[x] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

G.f. A(x) satisfies: A(x) = 1 + x * A(x)^2 + 7 * x^2 * A(x) * A'(x) + 6 * x^3 * A(x) * A''(x) + x^4 * A(x) * A'''(x).

cp's OEIS Frontend

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

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A256020 a(n) = Sum_{i=1..n-1} (i^4 * a(i)), a(1)=1.

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A376096 a(0) = 1; a(n) = Sum_{k=0..n-1} (k+1)^3 * a(k) * a(n-k-1).

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