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 1-4 of 4 results.

A204064 G.f.: Sum_{n>=0} x^n * Product_{k=1..n} (k + n*x) / (1 + k*x + n*x^2).

Original entry on oeis.org

1, 1, 2, 5, 14, 44, 152, 572, 2324, 10124, 47012, 231572, 1204964, 6599444, 37924292, 228033332, 1431128804, 9354072404, 63548071172, 447923400692, 3270361265444, 24696229801364, 192625876675652, 1549890430643252, 12849460733123684, 109647468132256724
Offset: 0

Views

Author

Paul D. Hanna, Jan 09 2013

Keywords

Comments

Compare to the identity:
Sum_{n>=0} x^n * Product_{k=1..n} (k + x) / (1 + k*x + x^2) = (1+x^2)/(1-x).
Compare to the g.f. of A187741:
Sum_{n>=0} x^n * (1 + n*x)^n / (1 + x + n*x^2)^n = 1/2 + (1+2*x)*Sum_{n>=0} (n+1)!*x^(2*n)/2.

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 44*x^5 + 152*x^6 + 572*x^7 +...
where
A(x) = 1 + x*(1+x)/(1+x+x^2) + x^2*(1+2*x)*(2+2*x)/((1+x+2*x^2)*(1+2*x+2*x^2)) + x^3*(1+3*x)*(2+3*x)*(3+3*x)/((1+x+3*x^2)*(1+2*x+3*x^2)*(1+3*x+3*x^2)) + x^4*(1+4*x)*(2+4*x)*(3+4*x)*(4+4*x)/((1+x+4*x^2)*(1+2*x+4*x^2)*(1+3*x+4*x^2)*(1+4*x+4*x^2)) +...
Also, we have the identity (cf. A229046):
A(x) = 1/2 + (1/2)*(1+x)/(1+x) + (2!/2)*x*(1+x)^2/((1+x)*(1+2*x)) + (3!/2)*x^2*(1+x)^3/((1+x)*(1+2*x)*(1+3*x)) + (4!/2)*x^3*(1+x)^4/((1+x)*(1+2*x)*(1+3*x)*(1+4*x)) + (5!/2)*x^4*(1+x)^5/((1+x)*(1+2*x)*(1+3*x)*(1+4*x)*(1+5*x)) +...

Links

Crossrefs

Cf. A229046, A204066, A187741, A187742, A220181.

Programs

  • Maple
    b:= proc(n, k) option remember; `if`(n<1, 1, `if`(k>
      ceil(n/2), 0, add((k-j)*b(n-1-j, k-j), j=0..1)))
    end:
a:= n-> ceil(add(b(n+2, k), k=1..1+ceil(n/2))/2):
seq(a(n), n=0..25);  # Alois P. Heinz, Jan 26 2018
  • Mathematica
    b[n_, k_] := b[n, k] = If[n < 1, 1, If[k > Ceiling[n/2], 0, Sum[(k - j) b[n - 1 - j, k - j], {j, 0, 1}]]];
a[n_] := Ceiling[Sum[b[n + 2, k], {k, 1, 1 + Ceiling[n/2]}]/2];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jun 05 2018, after Alois P. Heinz *)
  • PARI
    {a(n)=polcoeff( sum(m=0, n, x^m*prod(k=1,m,(k+m*x)/(1+k*x+m*x^2 +x*O(x^n))) ), n)}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    {a(n)=polcoeff( 1/2 + sum(m=1, n+1, m!/2*x^(m-1)*(1+x)^m/prod(k=1, m, 1+k*x +x*O(x^n))), n)}
for(n=0,30,print1(a(n),", "))
  • PARI
    {a(n)=if(n<0,0,if(n<1,1,(1/2)*sum(k=0, floor((n+1)/2), sum(i=0, k, (-1)^i*binomial(k, i)*(k-i+1)^(n-k+1)))))} \\ Paul D. Hanna, Jul 13 2014
for(n=0, 30, print1(a(n), ", "))

Formula

G.f.: 1/2 + Sum_{n>=1} n!/2 * x^(n-1) * (1+x)^n / Product_{k=1..n} (1 + k*x). - Paul D. Hanna, Oct 27 2013
a(n) = A229046(n+1)/2 for n>0.
a(n) = (1/2)*Sum_{k=0..floor((n+1)/2)} Sum_{i=0..k} (-1)^i * C(k,i) * (k-i+1)^(n-k+1) for n>1. (See Paul Barry's formula in A105795). - Paul D. Hanna, Jul 13 2014

A208237 G.f.: Sum_{n>=0} n! * x^n * Product_{k=1..n} (1 + k*x) / (1 + k*x + k^2*x^2).

Original entry on oeis.org

1, 1, 2, 5, 15, 54, 223, 1045, 5474, 31685, 200895, 1384470, 10304431, 82376101, 703949762, 6403761365, 61784985615, 630180031734, 6775001385343, 76572619018165, 907658144193314, 11259399965148005, 145879271404693215, 1970471655222795990, 27702625497930064591
Offset: 0

Views

Author

Paul D. Hanna, Jan 11 2013

Keywords

Comments

Compare to the identity:
Sum_{n>=0} n! * x^n * Product_{k=1..n} (1 + x) / (1 + k*x + k*x^2) = 1/(1-x-x^2).
Compare also to the g.f. of A136127:
x*Sum_{n>=0} n! * x^n * Product_{k=1..n} (2 + k*x) / (1 + 2*k*x + k^2*x^2).

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 15*x^4 + 54*x^5 + 223*x^6 + 1045*x^7 +...
where
A(x) = 1 + x*(1+x)/(1+x+x^2) + 2!*x^2*(1+x)*(1+2*x)/((1+x+x^2)*(1+2*x+4*x^2)) + 3!*x^3*(1+x)*(1+2*x)*(1+3*x)/((1+x+x^2)*(1+2*x+4*x^2)*(1+3*x+9*x^2)) + 4!*x^4*(1+x)*(1+2*x)*(1+3*x)*(1+4*x)/((1+x+x^2)*(1+2*x+4*x^2)*(1+3*x+9*x^2)*(1+4*x+16*x^2)) +...

Links

Crossrefs

Cf. A221370, A204064, A204066, A208236, A136127.

Programs

  • PARI
    {a(n)=polcoeff( sum(m=0, n, m!*x^m*prod(k=1, m, (1+k*x)/(1+k*x+k^2*x^2 +x*O(x^n))) ), n)}
for(n=0, 30, print1(a(n), ", "))

Formula

a(n) ~ 2 * 3^(n/2 + 5/4) * n^(n+2) / (exp(n) * Pi^(n+3/2)). - Vaclav Kotesovec, Nov 02 2014

A208236 G.f.: Sum_{n>=0} x^n * Product_{k=1..n} (1 + n*k*x) / (1 + x + n*k*x^2).

Original entry on oeis.org

1, 1, 1, 4, 10, 50, 208, 1290, 7456, 55982, 411796, 3650514, 32484460, 332970374, 3468625588, 40420787250, 481757564956, 6295577910182, 84407459209876, 1223095585594674, 18208380720893980, 289843786627539014, 4741844351895315028, 82269590167564595250
Offset: 0

Views

Author

Paul D. Hanna, Jan 10 2013

Keywords

Comments

Compare to the identity:
Sum_{n>=0} x^n * Product_{k=1..n} (1 + t*k*x) / (1 + x + t*k*x^2) = (1+x)/(1-t*x^2).
Compare to the g.f. of A187741:
Sum_{n>=0} x^n * (1 + n*x)^n / (1 + x + n*x^2)^n = 1/2 + (1+2*x)*Sum_{n>=0} (n+1)!*x^(2*n)/2.
Limit n->infinity (a(n)/n!)^(1/n) = 1/(2*log(2)). - Vaclav Kotesovec, Nov 03 2014

Examples

			G.f.: A(x) = 1 + x + x^2 + 4*x^3 + 10*x^4 + 50*x^5 + 208*x^6 + 1290*x^7 +...
where
A(x) = 1 + x*(1+x)/(1+x+x^2) + x^2*(1+2*1*x)*(1+2*2*x)/((1+x+2*1*x^2)*(1+x+2*2*x^2)) + x^3*(1+3*1*x)*(1+3*2*x)*(1+3*3*x)/((1+x+3*1*x^2)*(1+x+3*2*x^2)*(1+x+3*3*x^2)) + x^4*(1+4*1*x)*(1+4*2*x)*(1+4*3*x)*(1+4*4*x)/((1+x+4*1*x^2)*(1+x+4*2*x^2)*(1+x+4*3*x^2)*(1+x+4*4*x^2)) +...

Links

Crossrefs

Cf. A204064, A187741, A204066.

Programs

  • PARI
    {a(n)=polcoeff( sum(m=0, n, x^m*prod(k=1, m, (1+m*k*x)/(1+x+m*k*x^2 +x*O(x^n))) ), n)}
for(n=0, 30, print1(a(n), ", "))

A363110 G.f.: Sum_{n>=0} x^n * Product_{k=1..n} (k + (n-k+1)*x) / (1 + k*x + (n-k+1)*x^2).

Original entry on oeis.org

1, 1, 2, 4, 10, 28, 88, 306, 1158, 4730, 20722, 96776, 479340, 2507510, 13804014, 79718782, 481614806, 3036358968, 19932689952, 135981543762, 962319171782, 7053068549250, 53458038451082, 418440466421960, 3378290373259300, 28099682071640734, 240537280709926718
Offset: 0

Views

Author

Paul D. Hanna, Jun 02 2023

Keywords

Comments

Compare to the following identities, which hold for any fixed b and c:
(1) Sum_{n>=0} x^n * Product_{k=1..n} (b + k*x)/(1 + b*x + k*x^2) = (1 + b*x)/(1 - x^2).
(2) Sum_{n>=0} x^n * Product_{k=1..n} (k + c*x)/(1 + k*x + c*x^2) = (1 + c*x^2)/(1 - x).
(3) Sum_{n>=0} x^n * Product_{k=1..n} (b*k + c*k*x)/(1 + b*k*x + c*k*x^2) = 1/(1 - b*x - c*x^2).
Conjectures:
(1) a(6*n + k) == 0 (mod 4) for n > 0 when k = {0,5},
(2) a(6*n + k) == 2 (mod 4) for n > 0 when k = {1,2,3,4}.

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 10*x^4 + 28*x^5 + 88*x^6 + 306*x^7 + 1158*x^8 + 4730*x^9 + 20722*x^10 + 96776*x^11 + 479340*x^12 + ...
where
A(x) = 1 + x*(1+x)/(1+x+x^2) + x^2*(1 + 2*x)*(2 + x)/((1 + x + 2*x^2)*(1 + 2*x + x^2)) + x^3*(1 + 3*x)*(2 + 2*x)*(3 + x)/((1 + x + 3*x^2)*(1 + 2*x + 2*x^2)*(1 + 3*x + x^2)) + x^4*(1 + 4*x)*(2 + 3*x)*(3 + 2*x)*(4 + x)/((1 + x + 4*x^2)*(1 + 2*x + 3*x^2)*(1 + 3*x + 2*x^2)*(1 + 4*x + x^2)) + ...

Links

Crossrefs

Cf. A204064, A204066, A316370, A067948.

Programs

  • PARI
    {a(n) = polcoeff( A = sum(m=0, n, x^m*prod(k=1, m, (k + (m-k+1)*x)/(1 + k*x + (m-k+1)*x^2 +x*O(x^n))) ), n)}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n may be described by the following.
(1) Sum_{n>=0} x^n * Product_{k=1..n} (k + (n-k+1)*x) / (1 + k*x + (n-k+1)*x^2).
(2) Sum_{n>=0} x^n * (Sum_{k=0..n} A067948(n,k) * x^k) / Product_{k=1..n} (1 + k*x + (n-k+1)*x^2).
Showing 1-4 of 4 results.

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