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 1-9 of 9 results.

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!.

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!.

A075921 Second column of triangle A075502.

Original entry on oeis.org

1, 21, 343, 5145, 74431, 1058841, 14941423, 210003465, 2945813311, 41281739961, 578226834703, 8097153012585, 113373983463391, 1587332657497881, 22223335428043183, 311131443554114505
Offset: 0

Views

Author

Wolfdieter Lang, Oct 02 2002

Keywords

Comments

The e.g.f. given below is Sum_{m=0..1} A075513(2,m)*exp(7*(m+1)*x).

Links

Crossrefs

Cf. A000420 (first column), A075922.

Programs

  • Mathematica
    Table[-7^n+2 14^n,{n,0,20}] (* or *) LinearRecurrence[{21,-98}, {1,21},20] (* Harvey P. Dale, Apr 30 2011 *)

Formula

a(n) = A075502(n+2, 2) = (7^n)*S2(n+2, 2) with S2(n, m) := A008277(n, m) (Stirling2).
a(n) = -7^n + 2*14^n.
G.f.: 1/((1-7*x)*(1-14*x)).
E.g.f.: (d^2/dx^2)(((exp(7*x)-1)/7)^2)/2! = -exp(7*x) + 2*exp(14*x).
a(0)=1, a(1)=21, a(n) = 21a(n-1) - 98a(n-2). - Harvey P. Dale, Apr 30 2011

A075922 Third column of triangle A075502.

Original entry on oeis.org

1, 42, 1225, 30870, 722701, 16235562, 355888225, 7683656190, 164302593301, 3491636199282, 73902587019625, 1560051480424710, 32874455072382301, 691950889177526202, 14553192008156093425, 305928163614832076430
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(7*(m+1)*x)/2!.

Crossrefs

Cf. A075921, A075923.

Formula

a(n) = A075502(n+3, 3) = (7^n)*S2(n+3, 3) with S2(n, m) := A008277(n, m) (Stirling2).
a(n) = (7^n - 8*14^n + 9*21^n)/2.
G.f.: 1/Product_{k=1..3} (1 - 7*k*x).
E.g.f.: (d^3/dx^3)(((exp(7*x)-1)/7)^3)/3! = (exp(7*x) - 8*exp(14*x) + 9*exp(21*x))/2!.

A075923 Fourth column of triangle A075502.

Original entry on oeis.org

1, 70, 3185, 120050, 4084101, 130590390, 4012419145, 120031392250, 3525181576301, 102196720335710, 2935410756419505, 83751552660170850, 2377917929557166101, 67273652916778177030, 1898215473677945050265
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(7*(m+1)*x)/3!.

Crossrefs

Cf. A075922, A075924.

Formula

a(n) = A075502(n+4, 4) = (7^n)*S2(n+4, 4) with S2(n, m) := A008277(n, m) (Stirling2).
a(n) = (-7^n + 24*14^n - 81*21^n + 64*28^n)/3!.
G.f.: 1/Product_{k=1..4} (1 - 7*k*x).
E.g.f.: (d^4/dx^4)(((exp(7*x)-1)/7)^4)/4! = (-exp(7*x) + 24*exp(14*x) - 81*exp(21*x) + 64*exp(28*x))/3!.

A075501 Stirling2 triangle with scaled diagonals (powers of 6).

Original entry on oeis.org

1, 6, 1, 36, 18, 1, 216, 252, 36, 1, 1296, 3240, 900, 60, 1, 7776, 40176, 19440, 2340, 90, 1, 46656, 489888, 390096, 75600, 5040, 126, 1, 279936, 5925312, 7511616, 2204496, 226800, 9576, 168, 1, 1679616, 71383680
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(6*z) - 1)*x/6) - 1.

Examples

			[1]; [6,1]; [36,18,1]; ...; p(3,x) = x(36 + 18*x + x^2).
From _Andrew Howroyd_, Mar 25 2017: (Start)
Triangle starts
*      1
*      6       1
*     36      18       1
*    216     252      36       1
*   1296    3240     900      60      1
*   7776   40176   19440    2340     90    1
*  46656  489888  390096   75600   5040  126   1
* 279936 5925312 7511616 2204496 226800 9576 168 1
(End)

Links

Crossrefs

Columns 1-7 are A000400, A016175, A075916-A075920. Row sums are A005012.
Cf. A075500, A075502.

Programs

  • Maple
    # The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> 6^n, 9); # Peter Luschny, Jan 28 2016
  • Mathematica
    Flatten[Table[6^(n - m) StirlingS2[n, m], {n, 11}, {m, n}]] (* Indranil Ghosh, Mar 25 2017 *)
BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 10;
M = BellMatrix[6^#&, rows];
Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 23 2018, after Peter Luschny *)
  • PARI
    for(n=1, 11, for(m=1, n, print1(6^(n - m) * stirling(n, m, 2),", ");); print();) \\ Indranil Ghosh, Mar 25 2017

Formula

a(n, m) = (6^(n-m)) * stirling2(n, m).
a(n, m) = (Sum_{p=0..m-1} A075513(m, p)*((p+1)*6)^(n-m))/(m-1)! for n >= m >= 1, else 0.
a(n, m) = 6m*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}(1-6k*x), m >= 1.
E.g.f. for m-th column: (((exp(6x)-1)/6)^m)/m!, m >= 1.

A075503 Stirling2 triangle with scaled diagonals (powers of 8).

Original entry on oeis.org

1, 8, 1, 64, 24, 1, 512, 448, 48, 1, 4096, 7680, 1600, 80, 1, 32768, 126976, 46080, 4160, 120, 1, 262144, 2064384, 1232896, 179200, 8960, 168, 1, 2097152, 33292288, 31653888, 6967296, 537600, 17024, 224, 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(8*z) - 1)*x/8) - 1.

Examples

			[1]; [8,1]; [64,24,1]; ...; p(3,x) = x(64 + 24*x + x^2).
From _Andrew Howroyd_, Mar 25 2017: (Start)
Triangle starts
*       1
*       8        1
*      64       24        1
*     512      448       48       1
*    4096     7680     1600      80      1
*   32768   126976    46080    4160    120     1
*  262144  2064384  1232896  179200   8960   168   1
* 2097152 33292288 31653888 6967296 537600 17024 224 1
(End)

Links

Crossrefs

Columns 1-7 are A001018, A060195, A076003-A076007. Row sums are A075507.
Cf. A075502, A075504.

Programs

  • Mathematica
    Flatten[Table[8^(n - m) StirlingS2[n, m], {n, 11}, {m, n}]] (* Indranil Ghosh, Mar 25 2017 *)
  • PARI
    for(n=1, 11, for(m=1, n, print1(8^(n - m) * stirling(n, m, 2),", ");); print();) \\ Indranil Ghosh, Mar 25 2017

Formula

a(n, m) = (8^(n-m)) * stirling2(n, m).
a(n, m) = (Sum_{p=0..m-1} A075513(m, p)*((p+1)*8)^(n-m))/(m-1)! for n >= m >= 1, else 0.
a(n, m) = 8m*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}(1-8k*x), m >= 1.
E.g.f. for m-th column: (((exp(8x)-1)/8)^m)/m!, m >= 1.

A075506 Shifts one place left under 7th-order binomial transform.

Original entry on oeis.org

1, 1, 8, 71, 729, 8842, 125399, 2026249, 36458010, 719866701, 15453821461, 358100141148, 8899677678109, 235877034446341, 6634976621814472, 197269776623577659, 6177654735731310917, 203136983117907790890, 6994626418539177737803, 251584328242318030774781
Offset: 0

Views

Author

Wolfdieter Lang, Oct 02 2002

Keywords

Comments

Previous name was: Row sums of triangle A075502 (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->7^(n-m)*Stirling2(n,m))); # Muniru A Asiru, Mar 20 2018
  • Maple
    [seq(factorial(k)*coeftayl(exp((exp(7*x)-1)/7), x = 0, k), k=0..20)]; # Muniru A Asiru, Mar 20 2018
  • Mathematica
    Table[7^n BellB[n, 1/7], {n, 0, 20}]

Formula

a(n) = sum((7^(n-m))*S2(n,m), m=0..n), with S2(n,m) = A008277(n,m) (Stirling2).
E.g.f.: exp((exp(7*x)-1)/7).
O.g.f.: Sum_{k>=0} x^k/Product_{j=1..k} (1 - 7*j*x). - Ilya Gutkovskiy, Mar 20 2018
a(n) ~ 7^n * n^n * exp(n/LambertW(7*n) - 1/7 - n) / (sqrt(1 + LambertW(7*n)) * LambertW(7*n)^n). - Vaclav Kotesovec, Jul 15 2021

Extensions

a(0)=1 inserted and new name by Vladimir Reshetnikov, Oct 20 2015
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