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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A075925 Sixth column of triangle A075502.

Original entry on oeis.org

1, 147, 13034, 907578, 54807627, 3016638009, 155726334148, 7676501248416, 365698066506773, 16976491006185711, 772549060467762942, 34614587429584922214, 1532054031119984651839, 67151990527665760714053
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(7*(m+1)*x)/5!.

Crossrefs

Cf. A075924, A076002.

Programs

  • Mathematica
    CoefficientList[Series[1/Product[1-7k x,{k,6}],{x,0,20}],x] (* Harvey P. Dale, May 25 2012 *)

Formula

a(n) = A075502(n+6, 6) = (7^n)*S2(n+6, 6) with S2(n, m) := A008277(n, m) (Stirling2).
a(n) = Sum_{m=0..5} A075513(6, m)*((m+1)*7)^n/5!.
G.f.: 1/Product_{k=1..6} (1 - 7*k*x).
E.g.f.: (d^6/dx^6)(((exp(7*x)-1)/7)^6)/6! = (-exp(7*x) + 160*exp(14*x) - 2430*exp(21*x) + 10240*exp(28*x) - 15625*exp(35*x) + 7776*exp(42*x))/5!.

A076002 Seventh column of triangle A075502.

Original entry on oeis.org

1, 196, 22638, 2016840, 153632787, 10544644572, 672413918176, 40624783239040, 2356312445219733, 132435800821952628, 7261903300743441714, 390447849166013566200, 20663998640254649395639
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(7*(m+1)*x)/6!.

Crossrefs

Cf. A075525.

Formula

a(n) = A075502(n+7, 7) = (7^n)*S2(n+7, 7) with S2(n, m) := A008277(n, m) (Stirling2).
a(n) = Sum_{m=0..6} A075513(7, m)*((m+1)*7)^n/6!.
G.f.: 1/Product_{k=1..7} (1 - 7*k*x).
E.g.f.: (d^7/dx^7)(((exp(7*x)-1)/7)^7)/7! = (exp(7*x) - 384*exp(14*x) + 10935*exp(21*x) - 81920*exp(28*x) + 234375*exp(35*x) - 279936*exp(42*x) + 117649*exp(49*x))/6!.

A076004 Fourth column of triangle A075503.

Original entry on oeis.org

1, 80, 4160, 179200, 6967296, 254607360, 8940421120, 305659904000, 10259284361216, 339910422691840, 11158051230842880, 363834840082022400, 11805930580539867136, 381715961976738283520, 12309283295632755261440
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(8*(m+1)*x)/3!.

Links

Crossrefs

Cf. A076003, A076005.

Programs

Formula

a(n) = A075503(n+4, 4) = (8^n)*S2(n+4, 4) with S2(n, m) := A008277(n, m) (Stirling2).
a(n) = Sum_{m=0..3} ((A075513(4, m)*(m+1)*8)^n)/3!.
G.f.: 1/Product_{k=1..4} (1 - 8*k*x).
E.g.f.: (d^4/dx^4)(((exp(8*x)-1)/8)^4)/4! = (-exp(8*x) + 24*exp(16*x) - 81*exp(24*x) + 64*exp(32*x))/3!.

A076005 Fifth column of triangle A075503.

Original entry on oeis.org

1, 120, 8960, 537600, 28471296, 1393459200, 64678789120, 2892811468800, 125971743113216, 5378780147220480, 226309257119662080, 9416205124868505600, 388454135575280091136, 15919881384987941928960
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(8*(m+1)*x)/4!.

Links

Crossrefs

Cf. A076004, A076006.

Programs

Formula

a(n) = A075503(n+5, 5) = (8^n)*S2(n+5, 5) with S2(n, m) := A008277(n, m) (Stirling2).
a(n) = Sum_{m=0..4} (A075513(5, m)*((m+1)*8)^n)/4!.
G.f.: 1/Product_{k=1..5} (1 - 8*k*x).
E.g.f.: (d^5/dx^5)(((exp(8*x)-1)/8)^5)/5! = (exp(8*x) - 64*exp(16*x) + 486*exp(24*x) - 1024*exp(32*x) + 625*exp(40*x))/4!.

A076006 Sixth column of triangle A075503.

Original entry on oeis.org

1, 168, 17024, 1354752, 93499392, 5881430016, 346987429888, 19548208103424, 1064285732077568, 56464495286943744, 2936605030892961792, 150373246607730671616, 7606369972746352328704, 381025640076812853706752
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(8*(m+1)*x))/5!.

Links

Crossrefs

Cf. A076005, A076007.

Programs

Formula

a(n) = A075503(n+6, 6) = (8^n)*S2(n+6, 6) with S2(n, m) := A008277(n, m) (Stirling2).
a(n) = Sum_{m=0..5} (A075513(6, m)*((m+1)*8)^n)/5!.
G.f.: 1/Product_{k=1..6} (1 - 8*k*x).
E.g.f.: (d^6/dx^6)(((exp(8*x)-1)/8)^6)/6! = (-exp(8*x) + 160*exp(16*x) - 2430*exp(24*x) + 10240*exp(32*x) - 15625*exp(40*x) + 7776*exp(48*x))/5!.

A076007 Seventh column of triangle A075503.

Original entry on oeis.org

1, 224, 29568, 3010560, 262090752, 20558512128, 1498264109056, 103450998210560, 6857541631868928, 440486826671603712, 27603867324502769664, 1696189816779885772800, 102592999712419955605504
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(8*(m+1)*x))/6!.

Links

Crossrefs

Cf. A076006.

Programs

Formula

a(n) = A075503(n+7, 7) = (8^n)*S2(n+7, 7) with S2(n, m) := A008277(n, m) (Stirling2).
a(n) = Sum_{m=0..6} (A075513(7, m)*((m+1)*8)^n)/6!.
G.f.: 1/Product_{k=1..7} (1 - 8*k*x).
E.g.f.: (d^7/dx^7)(((exp(8*x)-1)/8)^7)/7! = (exp(8*x) - 384*exp(16*x) + 10935*exp(24*x) - 81920*exp(32*x) + 234375*exp(40*x) - 279936*exp(48*x) + 117649*exp(56*x))/6!.

A076008 Second column of triangle A075504.

Original entry on oeis.org

1, 27, 567, 10935, 203391, 3720087, 67493007, 1219657095, 21996874431, 396331160247, 7137447668847, 128505439098855, 2313380333315871, 41643387865514007, 749603858371707087, 13493075341822822215
Offset: 0

Views

Author

Wolfdieter Lang, Oct 02 2002

Keywords

Comments

The e.g.f. given below is Sum_{m=0..1} (A075513(3,m)*exp(9*(m+1)*x)).

Links

Crossrefs

Cf. A001019, A076009.

Programs

  • Mathematica
    CoefficientList[Series[1/((1-9x)(1-18x)),{x,0,30}],x] (* or *) LinearRecurrence[{27,-162},{1,27},30] (* Harvey P. Dale, Dec 01 2015 *)

Formula

a(n) = A075504(n+2, 2) = (9^n)*S2(n+2, 2) with S2(n, m) := A008277(n, m) (Stirling2).
a(n) = -9^n + 2*18^n.
G.f.: 1/((1-9*x)*(1-18*x)).
E.g.f.: (d^2/dx^2)(((exp(9*x)-1)/9)^2)/2! = -exp(9*x) + 2*exp(18*x).
a(0)=1, a(1)=27, a(n) = 27*a(n-1) - 162*a(n-2). - Harvey P. Dale, Dec 01 2015

A076009 Third column of triangle A075504.

Original entry on oeis.org

1, 54, 2025, 65610, 1974861, 57041334, 1607609025, 44625100770, 1226874595221, 33521945231214, 912229968911625, 24758714599712730, 670798674525559581, 18153207600055622694, 490886209059873519825
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(9*(m+1)*x))/2!.

Links

Crossrefs

Cf. A076008, A076010.

Programs

Formula

a(n) = A075504(n+3, 3) = (9^n)*S2(n+3, 3) with S2(n, m) := A008277(n, m) (Stirling2).
a(n) = (9^n - 8*18^n + 9*27^n)/2.
G.f.: 1/Product_{k=1..3} (1 - 9*k*x).
E.g.f.: (d^3/dx^3)(((exp(9*x)-1)/9)^3)/3! = (exp(9*x) - 8*exp(18*x) + 9*exp(27*x))/2!.

A076010 Fourth column of triangle A075504.

Original entry on oeis.org

1, 90, 5265, 255150, 11160261, 458810730, 18124795305, 697117731750, 26323112938221, 981154011007170, 36233774365169745, 1329174591745823550, 48521083977375207381, 1764912230785563088410, 64027726517340144702585
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(9*(m+1)*x))/3!.

Links

Crossrefs

Cf. A076009, A076011.

Programs

Formula

a(n) = A075504(n+4, 4) = (9^n)*S2(n+4, 4) with S2(n, m) := A008277(n, m) (Stirling2).
a(n) = (-9^n + 24*18^n - 81*27^n + 64*36^n)/3!.
G.f.: 1/Product_{k=1..4} (1 - 9*k*x).
E.g.f.: (d^4/dx^4)(((exp(9*x)-1)/9)^4)/4! = (-exp(9*x) + 24*exp(18*x) - 81*exp(27*x) + 64*exp(36*x))/3!.

A076011 Fifth column of triangle A075504.

Original entry on oeis.org

1, 135, 11340, 765450, 45605511, 2511058725, 131122437930, 6597627438600, 323216347675221, 15525889656392115, 734898808902814920, 34399620992372494950, 1596504028634137480131, 73607593519321749694305
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(9*(m+1)*x))/4!.

Links

Crossrefs

Cf. A076010, A076012.

Programs

Formula

a(n) = A075504(n+5, 5) = (9^n)*S2(n+5, 5) with S2(n, m) := A008277(n, m) (Stirling2).
a(n) = Sum_{m=0..4} (A075513(5, m)*(9*(m+1))^n)/4!.
G.f.: 1/Product_{k=1..5} (1 - 9*k*x).
E.g.f.: (d^5/dx^5)(((exp(9*x)-1)/9)^5)/5! = (exp(9*x) - 64*exp(18*x) + 486*exp(27*x) - 1024*exp(36*x) + 625*exp(45*x))/4!.
Previous Showing 31-40 of 55 results. Next

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