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 1-8 of 8 results.

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!.

A076003 Third column of triangle A075503.

Original entry on oeis.org

1, 48, 1600, 46080, 1232896, 31653888, 792985600, 19566428160, 478167433216, 11613323132928, 280917704704000, 6777200695050240, 163215697915150336, 3926183399462535168, 94372512377130188800
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(8*(m+1)*x)/2!.

Crossrefs

Cf. A060195, A076004.

Formula

a(n) = A075503(n+3, 3) = (8^n)*S2(n+3, 3) with S2(n, m) := A008277(n, m) (Stirling2).
a(n) = (8^n - 8*16^n + 9*24^n)/2.
G.f.: 1/Product_{k=1..3} (1 - 8*k*x).
E.g.f.: (d^3/dx^3)(((exp(8*x)-1)/8)^3)/3! = (exp(8*x) - 8*exp(16*x) + 9*exp(24*x))/2!.

A075502 Triangle read by rows: Stirling2 triangle with scaled diagonals (powers of 7).

Original entry on oeis.org

1, 7, 1, 49, 21, 1, 343, 343, 42, 1, 2401, 5145, 1225, 70, 1, 16807, 74431, 30870, 3185, 105, 1, 117649, 1058841, 722701, 120050, 6860, 147, 1, 823543, 14941423, 16235562, 4084101, 360150, 13034, 196, 1
Offset: 1

Views

Author

Wolfdieter Lang, Oct 02 2002

Keywords

Comments

This is a lower triangular infinite matrix of the Jabotinsky type. See the D. E. Knuth reference given in A039692 for exponential convolution arrays.
The row polynomials p(n,x) := Sum_{m=1..n} a(n,m)x^m, n >= 1, have e.g.f. J(x; z)= exp((exp(7*z) - 1)*x/7) - 1.

Examples

			[1]; [7,1]; [49,21,1]; ...; p(3,x) = x * (49 + 21*x + x^2).
From _Andrew Howroyd_, Mar 25 2017: (Start)
Triangle starts
*      1
*      7        1
*     49       21        1
*    343      343       42       1
*   2401     5145     1225      70      1
*  16807    74431    30870    3185    105     1
* 117649  1058841   722701  120050   6860   147   1
* 823543 14941423 16235562 4084101 360150 13034 196 1
(End)

Links

Crossrefs

Columns 1-7 are A000420, A075921-A075925, A076002. Row sums are A075506.
Cf. A075501, A075503.

Programs

  • Mathematica
    Flatten[Table[7^(n - m) StirlingS2[n, m], {n, 11}, {m, n}]] (* Indranil Ghosh, Mar 25 2017 *)
  • PARI
    for(n=1, 11, for(m=1, n, print1(7^(n - m) * stirling(n, m, 2),", ");); print();) \\ Indranil Ghosh, Mar 25 2017

Formula

a(n, m) = (7^(n-m)) * stirling2(n, m).
a(n, m) = 7*m*a(n-1, m) + a(n-1, m-1), n>=m>=1, else 0, with a(n, 0) := 0 and a(1, 1)=1.
a(n, m) = (Sum_{p=0..m-1} A075513(m, p)*((p+1)*7)^(n-m))/(m-1)! for n >= m >= 1, else 0.
G.f. for m-th column: (x^m)/Product_{k=1..m}(1-7*k*x), m >= 1.
E.g.f. for m-th column: (((exp(7*x)-1)/7)^m)/m!, m >= 1.

A075504 Stirling2 triangle with scaled diagonals (powers of 9).

Original entry on oeis.org

1, 9, 1, 81, 27, 1, 729, 567, 54, 1, 6561, 10935, 2025, 90, 1, 59049, 203391, 65610, 5265, 135, 1, 531441, 3720087, 1974861, 255150, 11340, 189, 1, 4782969, 67493007, 57041334, 11160261, 765450, 21546, 252, 1
Offset: 1

Views

Author

Wolfdieter Lang, Oct 02 2002

Keywords

Comments

This is a lower triangular infinite matrix of the Jabotinsky type. See the Knuth reference given in A039692 for exponential convolution arrays.
The row polynomials p(n,x) := Sum_{m=1..n} a(n,m)x^m, n >= 1, have e.g.f. J(x; z)= exp((exp(9*z) - 1)*x/9) - 1.
Row sums give A075508(n), n >= 1. The columns (without leading zeros) give A001019 (powers of 9), A076008-A076013 for m=1..7.

Examples

			[1]; [9,1]; [81,27,1]; ...; p(3,x) = x(81 + 27*x + x^2).
From _Andrew Howroyd_, Mar 25 2017: (Start)
Triangle starts
*       1
*       9        1
*      81       27        1
*     729      567       54        1
*    6561    10935     2025       90      1
*   59049   203391    65610     5265    135     1
*  531441  3720087  1974861   255150  11340   189   1
* 4782969 67493007 57041334 11160261 765450 21546 252 1
(End)

Links

Crossrefs

Cf. A075503, A075505.
Columns 2-7 are A076008-A076013.

Programs

  • Mathematica
    Flatten[Table[9^(n - m) StirlingS2[n, m], {n, 11}, {m, n}]] (* Indranil Ghosh, Mar 25 2017 *)
  • PARI
    for(n=1, 11, for(m=1, n, print1(9^(n - m) * stirling(n, m, 2),", ");); print();) \\ Indranil Ghosh, Mar 25 2017

Formula

a(n, m) = (9^(n-m)) * stirling2(n, m).
a(n, m) = Sum_{p=0..m-1} (A075513(m, p)*((p+1)*9)^(n-m))/(m-1)! for n >= m >= 1, else 0.
a(n, m) = 9m*a(n-1, m) + a(n-1, m-1), n >= m >= 1, else 0, with a(n, 0) := 0 and a(1, 1)=1.
G.f. for m-th column: (x^m)/Product_{k=1..m}(1-9k*x), m >= 1.
E.g.f. for m-th column: (((exp(9x) - 1)/9)^m)/m!, m >= 1.

A075507 Shifts one place left under 8th-order binomial transform.

Original entry on oeis.org

1, 1, 9, 89, 1009, 13457, 210105, 3747753, 74565473, 1628999841, 38704241897, 993034281593, 27340167242321, 803154583649329, 25050853217628313, 826165199464341705, 28707262835597618369, 1047731789671001235265, 40053733152627299592137, 1599910554128824794493593
Offset: 0

Views

Author

Wolfdieter Lang, Oct 02 2002

Keywords

Comments

Previous name was: Row sums of triangle A075503 (for n>=1).

Links

Crossrefs

Shifts one place left under k-th order binomial transform, k=1..10: A000110, A004211, A004212, A004213, A005011, A005012, A075506, A075507, A075508, A075509.

Programs

  • GAP
    List([0..20],n->Sum([0..n],m->8^(n-m)*Stirling2(n,m))); # Muniru A Asiru, Mar 20 2018
  • Maple
    [seq(factorial(k)*coeftayl(exp((exp(8*x)-1)/8), x = 0, k), k=0..20)]; # Muniru A Asiru, Mar 20 2018
  • Mathematica
    Table[8^n BellB[n, 1/8], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 20 2015 *)

Formula

a(n) = Sum_{m=0..n} 8^(n-m)*S2(n,m), with S2(n,m) = A008277(n,m) (Stirling2).
E.g.f.: exp((exp(8*x)-1)/8).
O.g.f.: Sum_{k>=0} x^k/Product_{j=1..k} (1 - 8*j*x). - Ilya Gutkovskiy, Mar 20 2018
a(n) ~ 8^n * n^n * exp(n/LambertW(8*n) - 1/8 - n) / (sqrt(1 + LambertW(8*n)) * LambertW(8*n)^n). - Vaclav Kotesovec, Jul 15 2021

Extensions

a(0)=1 inserted and new name by Vladimir Reshetnikov, Oct 20 2015
Showing 1-8 of 8 results.

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