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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A076013 Seventh column of triangle A075504.

Original entry on oeis.org

1, 252, 37422, 4286520, 419818707, 37047106404, 3037410645984, 235940417032320, 17594974122819093, 1271468563282273356, 89638618747098243186, 6196581962116572990600, 421646012618644954061559
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(9*(m+1)*x))/6!.

Links

Crossrefs

Cf. A075504, A075513, A076012.

Programs

Formula

a(n) = A075504(n+7, 7) = (9^n)*S2(n+7, 7) with S2(n, m) := A008277(n, m) (Stirling2).
a(n) = Sum_{m=0..6} (A075513(7, m)*((m+1)*9)^n)/6!.
G.f.: 1/Product_{k=1..7} (1 - 9*k*x).
E.g.f.: (d^7/dx^7)(((exp(9*x)-1)/9)^7)/7! = (exp(9*x) - 384*exp(18*x) + 10935*exp(27*x) - 81920*exp(36*x) + 234375*exp(45*x) - 279936*exp(54*x) + 117649*exp(63*x))/6!.

A028085 Expansion of 1/((1-3x)(1-6x)(1-9x)(1-12x)).

Original entry on oeis.org

1, 30, 585, 9450, 137781, 1888110, 24862545, 318755250, 4012058061, 49847787990, 613622150505, 7503229474650, 91300979746341, 1106997911204670, 13386607046238465, 161563913916523650
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Fourth column of triangle A075498.
Cf. A017933, A075515.

Programs

  • Mathematica
    CoefficientList[Series[1/((1-3x)(1-6x)(1-9x)(1-12x)),{x,0,30}],x] (* or *) LinearRecurrence[{30,-315,1350,-1944},{1,30,585,9450},30] (* Harvey P. Dale, Feb 06 2015 *)
  • PARI
    Vec(1/((1-3*x)*(1-6*x)*(1-9*x)*(1-12*x))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012

Formula

a(n) = (3^n)*Stirling2(n+4, 4), n >= 0, with Stirling2(n, m) = A008277(n, m).
a(n) = Sum_{m=0..3} (A075513(4, m)*((m+1)*3)^n)/3!.
G.f.: 1/Product_{k=1..4} (1-3*k*x).
E.g.f.: (d^4/dx^4)((((exp(3*x)-1)/3)^4)/4!) = Sum_{m=0..3} (A075513(4, m)*exp(3*(m+1)*x))/3!.
a(n) = (12^(n+3) - 3*9^(n+3) + 3*6^(n+3) - 3^(n+3))/162. - Yahia Kahloune, Jun 10 2013
a(0)=1, a(1)=30, a(2)=585, a(3)=9450, a(n) = 30*a(n-1) - 315*a(n-2) + 1350*a(n-3) - 1944*a(n-4). - Harvey P. Dale, Feb 06 2015

A075511 Sixth column of triangle A075497.

Original entry on oeis.org

1, 42, 1064, 21168, 365232, 5743584, 84713728, 1193127936, 16239711488, 215394955776, 2800564795392, 35851775791104, 453374980255744, 5677724481773568, 70550796621971456, 871159544637161472
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(2*(m+1)*x)/5!.

Crossrefs

Cf. A075510, A075512.

Formula

a(n) = A075497(n+6, 6) = (2^n)*S2(n+6, 6) with S2(n, m) := A008277(n, m) (Stirling2).
a(n) = Sum_{m=0..5} A075513(6, m)*((m+1)*2)^n/5!.
G.f.: 1/Product_{k=1..6} (1 - 2*k*x).
E.g.f.: (d^6/dx^6)(((exp(2*x)-1)/2)^6)/6! = (-exp(2*x) + 160*exp(4*x) - 2430*exp(6*x) + 10240*exp(8*x) - 15625*exp(10*x) + 7776*exp(12*x))/5!.

A075512 Seventh column of triangle A075497.

Original entry on oeis.org

1, 56, 1848, 47040, 1023792, 20076672, 365787136, 6314147840, 104637781248, 1680323893248, 26325099300864, 404403166003200, 6115019304300544, 91287994741981184, 1348582723009708032
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(2*(m+1)*x)/6!.

Crossrefs

Cf. A075511.

Formula

a(n) = A075497(n+7, 7) = (2^n)*S2(n+7, 7) with S2(n, m) := A008277(n, m) (Stirling2).
a(n) = Sum_{m=0..6} A075513(7, m)*((m+1)*2)^n/6!.
G.f.: 1/Product_{k=1..7} (1 - 2*k*x).
E.g.f.: (d^7/dx^7)(((exp(2*x)-1)/2)^7)/7! = (exp(2*x) - 384*exp(4*x) + 10935*exp(6*x) - 81920*exp(8*x) + 234375*exp(10*x) - 279936*exp(12*x) + 117649*exp(14*x))/6!.

A075906 Seventh column of triangle A075498.

Original entry on oeis.org

1, 84, 4158, 158760, 5182947, 152457228, 4166544096, 107883135360, 2681751885813, 64597295294532, 1518037879508514, 34979886546859800, 793401360863472999, 17766424516726033596, 393690756719422620612
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(3*(m+1)*x)/6!.

Crossrefs

Cf. A075516.

Formula

a(n) = A075498(n+7, 7) = (3^n)*S2(n+7, 7) with S2(n, m) := A008277(n, m) (Stirling2).
a(n) = Sum_{m=0..6} A075513(7, m)*((m+1)*3)^n/6!.
G.f.: 1/Product_{k=1..7} (1 - 3*k*x).
E.g.f.: (d^7/dx^7)(((exp(3*x)-1)/3)^7)/7! = (exp(3*x) - 384*exp(6*x) + 10935*exp(9*x) - 81920*exp(12*x) + 234375*exp(15*x) - 279936*exp(18*x) + 117649*exp(21*x))/6!.

A075907 Fourth column of triangle A075499.

Original entry on oeis.org

1, 40, 1040, 22400, 435456, 7956480, 139694080, 2387968000, 40075329536, 663887544320, 10896534405120, 177653730508800, 2882307270639616, 46596186764738560, 751299029274460160, 12089975328525516800
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(4*(m+1)*x)/3!.

Links

Crossrefs

Cf. A019677, A075908.

Programs

  • Mathematica
    Table[(-4^n+24*8^n-81*12^n+64*16^n)/6,{n,0,20}] (* or *) LinearRecurrence[ {40,-560,3200,-6144},{1,40,1040,22400},20] (* Harvey P. Dale, Jun 04 2013 *)

Formula

a(n) = A075499(n+4, 4) = (4^n)*S2(n+4, 4) with S2(n, m) := A008277(n, m) (Stirling2).
a(n) = (-4^n + 24*8^n - 81*12^n + 64*16^n)/3!.
G.f.: 1/Product_{k=1..4} (1 - 4*k*x).
E.g.f.: (d^4/dx^4)(((exp(4*x)-1)/4)^4)/4! = (-exp(4*x) + 24*exp(8*x) - 81*exp(12*x) + 64*exp(16*x))/3!.
a(0)=1, a(1)=40, a(2)=1040, a(3)=22400, a(n) = 40*a(n-1) - 560*a(n-2) + 3200*a(n-3) - 6144*a(n-4). - Harvey P. Dale, Jun 04 2013

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

A075916 Third column of triangle A075501.

Original entry on oeis.org

1, 36, 900, 19440, 390096, 7511616, 141134400, 2611802880, 47870735616, 871982724096, 15819463296000, 286235993272320, 5170077903015936, 93275375604350976, 1681524519443251200, 30298254922942709760
Offset: 0

Views

Author

Wolfdieter Lang, Oct 02 2002

Keywords

Comments

The e.g.f. given below is Sum_{m=0..2} A075513(3,m)*exp(6*(m+1)*x)/2!.

Crossrefs

Cf. A016175, A075917.

Formula

a(n) = A075501(n+3, 3) = (6^n)*S2(n+3, 3) with S2(n, m) := A008277(n, m) (Stirling2).
a(n) = (6^n - 8*12^n + 9*18^n)/2.
G.f.: 1/Product_{k=1..3} (1 - 6*k*x).
E.g.f.: (d^3/dx^3)(((exp(6*x)-1)/6)^3)/3! = (exp(6*x) - 8*exp(12*x) + 9*exp(18*x))/2!.

A075924 Fifth column of triangle A075502.

Original entry on oeis.org

1, 105, 6860, 360150, 16689351, 714717675, 29027537770, 1135995214200, 43285014073301, 1617172212901245, 59536438207963080, 2167526889938878650, 78241359077417918851, 2805721220626405336815, 100098458195602131838790
Offset: 0

Views

Author

Wolfdieter Lang, Oct 02 2002

Keywords

Comments

The e.g.f. given below is Sum_{m=0..4} A075513(5,m)*exp(7*(m+1)*x)/4!.

Crossrefs

Cf. A075923, A075925.

Formula

a(n) = A075502(n+5, 5) = (7^n)*S2(n+5, 5) with S2(n, m) := A008277(n, m) (Stirling2).
a(n) = Sum_{m=0..4} A075513(5, m)*((m+1)*7)^n/4!.
G.f.: 1/Product_{k=1..5} (1 - 7*k*x).
E.g.f.: (d^5/dx^5)(((exp(7*x)-1)/7)^5)/5! = (exp(7*x) - 64*exp(14*x) + 486*exp(21*x) - 1024*exp(28*x) + 625*exp(35*x))/4!.
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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