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.

A075912 Fourth column of triangle A075500.

Original entry on oeis.org

1, 50, 1625, 43750, 1063125, 24281250, 532890625, 11386718750, 238867578125, 4946347656250, 101481884765625, 2068161621093750, 41943091064453125, 847579699707031250, 17082562164306640625, 343617765808105468750, 6901873153839111328125
Offset: 0

Views

Author

Wolfdieter Lang, Oct 02 2002

Keywords

Comments

The e.g.f. given below is (Sum_{m=0..3} A075513(4,m)*exp(5*(m+1)*x))/3!.

Links

Crossrefs

Cf. A075911, A075913.

Programs

  • Mathematica
    Table[5^n*(-1 + 3*2^(3+n) + 2^(6+2*n) - 3^(4+n))/6, {n, 0, 20}] (* Vaclav Kotesovec, Dec 12 2015 *)
  • PARI
    Vec(1/((1-5*x)*(1-10*x)*(1-15*x)*(1-20*x)) + O(x^30)) \\ Colin Barker, Dec 11 2015

Formula

a(n) = A075500(n+4, 4) = (5^n)*S2(n+4, 4) with S2(n, m) = A008277(n, m) (Stirling2).
a(n) = (Sum_{m=0..3}A075513(4, m)*((m+1)*5)^n, m=0..3)/3!.
G.f.: 1/Product_{k=1..4}(1-5*k*x).
E.g.f.: (d^4/dx^4)((((exp(5*x)-1)/5)^4)/4!) = (-exp(5*x) + 24*exp(10*x) - 81*exp(15*x) + 64*exp(20*x))/3!.
a(n) = 50*a(n-1) - 875*a(n-2) + 6250*a(n-3) - 15000*a(n-4) for n>3. - Colin Barker, Dec 11 2015

A075914 Sixth column of triangle A075500.

Original entry on oeis.org

1, 105, 6650, 330750, 14266875, 560896875, 20682062500, 728227500000, 24779833203125, 821666548828125, 26708267167968750, 854772944238281250, 27023254648193359375, 846046877171630859375, 26282219820458984375000, 811330550012329101562500
Offset: 0

Views

Author

Wolfdieter Lang, Oct 02 2002

Keywords

Comments

The e.g.f. given below is Sum_{m=0..5}(A075513(6,m)*exp(5*(m+1)*x))/5!.

Links

Crossrefs

Cf. A000351, A016164, A075911, A075912, A075913, A075915.

Programs

  • Mathematica
    Table[5^(n-1) * (-1 + 5*2^(5+n) + 5*2^(11+2*n) - 10*3^(5+n) - 5^(6+n) + 6^(5+n))/24, {n, 0, 20}] (* Vaclav Kotesovec, Dec 12 2015 *)
  • PARI
    Vec(1/((1-5*x)*(1-10*x)*(1-15*x)*(1-20*x)*(1-25*x)*(1-30*x)) + O(x^30)) \\ Colin Barker, Dec 12 2015

Formula

a(n) = A075500(n+6, 6) = (5^n)*S2(n+6, 6) with S2(n, m) = A008277(n, m) (Stirling2).
a(n) = Sum_{m=0..5}(A075513(6, m)*((m+1)*5)^n)/5!.
G.f.: 1/Product_{k=1..6}(1-5*k*x).
E.g.f.: (d^6/dx^6)((((exp(5*x)-1)/5)^6)/6!) = (-exp(5*x) + 160*exp(10*x) - 2430*exp(15*x) + 10240*exp(20*x) - 15625*exp(25*x) + 7776*exp(30*x))/5!.
G.f.: 1 / ((1-5*x)*(1-10*x)*(1-15*x)*(1-20*x)*(1-25*x)*(1-30*x)). - Colin Barker, Dec 12 2015

A075915 Seventh column of triangle A075500.

Original entry on oeis.org

1, 140, 11550, 735000, 39991875, 1960612500, 89303500000, 3853850000000, 159664583203125, 6409926960937500, 251055710800781250, 9641722822265625000, 364483553427490234375, 13602971247133789062500, 502386213470141601562500, 18394848021467285156250000
Offset: 0

Views

Author

Wolfdieter Lang, Oct 02 2002

Keywords

Comments

The e.g.f. given below is Sum_{m=0..6}(A075513(7,m)exp(5*(m+1)*x))/6!.

Links

Crossrefs

Cf. A000351, A016164, A075911, A075912, A075913, A075914.

Programs

  • Mathematica
    Table[5^(n-1) * (1 - 3*2^(7 + n) - 5*2^(14 + 2*n) + 5*3^(7 + n) + 3*5^(7 + n) - 6^(7 + n) + 7^(6 + n))/144, {n, 0, 20}] (* Vaclav Kotesovec, Dec 12 2015 *)
  • PARI
    Vec(1/((1-5*x)*(1-10*x)*(1-15*x)*(1-20*x)*(1-25*x)*(1-30*x)*(1-35*x)) + O(x^30)) \\ Colin Barker, Dec 12 2015

Formula

a(n) = A075500(n+7, 7) = (5^n)S2(n+7, 7) with S2(n, m) = A008277(n, m) (Stirling2).
a(n) = Sum_{m=0..6}(A075513(7, m)*(5*(m+1))^n)/6!.
G.f.: 1/Product_{k=1..7}(1-5k*x).
E.g.f.: (d^7/dx^7)((((exp(5x)-1)/5)^7)/7!) = (exp(5*x) - 384*exp(10*x) + 10935*exp(15*x) - 81920*exp(20*x) + 234375*exp(25*x) - 279936*exp(30*x) + 117649*exp(35*x))/6!.
G.f.: 1 / ((1-5*x)*(1-10*x)*(1-15*x)*(1-20*x)*(1-25*x)*(1-30*x)*(1-35*x)). - Colin Barker, Dec 12 2015
Showing 1-3 of 3 results.

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

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