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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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.

A075911 Third column of triangle A075500.

Original entry on oeis.org

1, 30, 625, 11250, 188125, 3018750, 47265625, 728906250, 11133203125, 168996093750, 2554931640625, 38523925781250, 579858642578125, 8717878417968750, 130968170166015625, 1966522521972656250, 29517837677001953125, 442967564392089843750, 6646513462066650390625
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(5*(m+1)*x)/2!.

Links

Crossrefs

Cf. A016164, A075912.

Programs

  • Mathematica
    LinearRecurrence[{30,-275,750},{1,30,625},30] (* Harvey P. Dale, Oct 06 2017 *)
  • PARI
    Vec(1/((1-5*x)*(1-10*x)*(1-15*x)) + O(x^30)) \\ Colin Barker, Dec 11 2015

Formula

a(n) = A075500(n+3, 3) = (5^n)*S2(n+3, 3) with S2(n, m) = A008277(n, m) (Stirling2).
a(n) = (5^n - 8*10^n + 9*15^n)/2.
G.f.: 1/Product_{k=1..3} (1 - 5*k*x).
E.g.f.: (d^3/dx^3)(((exp(5*x)-1)/5)^3)/3! = (exp(5*x) - 8*exp(10*x) + 9*exp(15*x))/2!.
a(n) = 30*a(n-1) - 275*a(n-2) + 750*a(n-3) for n > 2. - Colin Barker, Dec 11 2015

A075515 Fifth column of triangle A075498.

Original entry on oeis.org

1, 45, 1260, 28350, 563031, 10333575, 179866170, 3016747800, 49263275061, 788796913905, 12445575859080, 194186867360850, 3004103990159091, 46168557763591035, 705914973500103990, 10750288516418083500
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(3*(m+1)*x)/4!.

Crossrefs

Cf. A028085, A075516.

Formula

a(n) = A075498(n+5, 5) = (3^n)*S2(n+5, 5) with S2(n, m) := A008277(n, m) (Stirling2).
a(n) = Sum_{m=0..4} A075513(5, m)*exp((m+1)*3)^n/4!.
G.f.: 1/Product_{k=1..5} (1 - 3*k*x).
E.g.f.: (d^5/dx^5)(((exp(3*x)-1)/3)^5)/5! = (exp(3*x) - 64*exp(6*x) + 486*exp(9*x) - 1024*exp(12*x) + 625*exp(15*x))/4!.

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

A258773 Triangle read by rows, T(n,k) = (-1)^(n-k)*C(n,k)*k^n, for n >= 0 and 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, -2, 4, 0, 3, -24, 27, 0, -4, 96, -324, 256, 0, 5, -320, 2430, -5120, 3125, 0, -6, 960, -14580, 61440, -93750, 46656, 0, 7, -2688, 76545, -573440, 1640625, -1959552, 823543, 0, -8, 7168, -367416, 4587520, -21875000, 47029248, -46118408, 16777216
Offset: 0

Views

Author

Peter Luschny, Jun 09 2015

Keywords

Comments

The row polynomials are p(0, x) = 1, and p(n, x) = Eu(x)^n (x-1)^n, for n >= 1, where Eu(x) := x*d/dx is the Euler derivative with respect to x. See A075513. - Wolfdieter Lang, Oct 12 2022
Coefficients of the Sidi polynomials (-1)^n*x*D_{n,0,n}(x), for n >= 0, where D_{k,n,m}(z) is given in Theorem 4.2., p. 862, of Sidi [1980]. - Wolfdieter Lang, Apr 10 2023

Examples

			Triangle begins:
  [1]
  [0,  1]
  [0, -2,     4]
  [0,  3,   -24,     27]
  [0, -4,    96,   -324,     256]
  [0,  5,  -320,   2430,   -5120,    3125]
  [0, -6,   960, -14580,   61440,  -93750,    46656]
  [0,  7, -2688,  76545, -573440, 1640625, -1959552, 823543]

Links

Crossrefs

Cf. A000142, A000312, A072034, A075513, A154715.

Programs

  • Maple
    seq(seq((-1)^(n-k)*binomial(n, k)*k^n, k=0..n), n=0..8);
T_row := proc(n) (-1)^n*(1-exp(x))^n/n!; diff(%,[x$n]); subs(exp(x)=t, n!*expand(%,x)); CoefficientList(%,t) end: seq(print(T_row(n)), n=0..7);
  • Mathematica
    Flatten@Table[(-1)^(n - k) Binomial[n, k] k^n, {n, 0 , 10}, {k, 0, n}] (* G. C. Greubel, Dec 17 2015 *)

Formula

Sum_{k=0..n} T(n,k) = n!.
Sum_{k=0..n} |T(n,k)| = A072034(n).
Sum_{n>=0} Sum_{k=0..n} T(n,k) x^k y^n/n! = 1/(1 + W(-x*y*exp(-y))) where W is the Lambert W function. - Robert Israel, Dec 16 2015
T(n,n) = A000312(n). - Peter Luschny, Dec 17 2015
T(n, k+1) = n * A075513(n, k) if n > 0. - Michael Somos, May 13 2018

A075505 Stirling2 triangle with scaled diagonals (powers of 10).

Original entry on oeis.org

1, 10, 1, 100, 30, 1, 1000, 700, 60, 1, 10000, 15000, 2500, 100, 1, 100000, 310000, 90000, 6500, 150, 1, 1000000, 6300000, 3010000, 350000, 14000, 210, 1, 10000000, 127000000, 96600000, 17010000, 1050000, 26600, 280, 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(10*z) - 1)*x/10) - 1.

Examples

			[1]; [10,1]; [100,30,1]; ...; p(3,x) = x(100 + 30*x + x^2).
From _Andrew Howroyd_, Mar 25 2017: (Start)
Triangle starts
*        1
*       10         1
*      100        30        1
*     1000       700       60        1
*    10000     15000     2500      100       1
*   100000    310000    90000     6500     150     1
*  1000000   6300000  3010000   350000   14000   210   1
* 10000000 127000000 96600000 17010000 1050000 26600 280 1
(End)

Links

Crossrefs

Row sums are A075509.
Cf. A075504.

Programs

  • Mathematica
    Flatten[Table[10^(n - m) StirlingS2[n, m], {n, 11}, {m, n}]] (* Indranil Ghosh, Mar 25 2017 *)
  • PARI
    for(n=1, 11, for(m=1, n, print1(10^(n - m) * stirling(n, m, 2),", ");); print();) \\ Indranil Ghosh, Mar 25 2017

Formula

a(n, m) = (10^(n-m)) * stirling2(n, m).
a(n, m) = (Sum_{p=0..m-1} A075513(m, p)*((p+1)*10)^(n-m))/(m-1)! for n >= m >= 1, else 0.
a(n, m) = 10m*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} (1-10k*x), m >= 1.
E.g.f. for m-th column: (((exp(10x)-1)/10)^m)/m!, m >= 1.

A075510 Fifth column of triangle A075497.

Original entry on oeis.org

1, 30, 560, 8400, 111216, 1360800, 15790720, 176563200, 1922176256, 20518417920, 215825326080, 2244998246400, 23153670762496, 237224718704640, 2418102840688640, 24549985173504000, 248464183682727936
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(2*(m+1)*x)/4!.

Crossrefs

Cf. A025966, A075511.

Formula

a(n) = A075497(n+5, 5) = (2^n)*S2(n+5, 5) with S2(n, m) := A008277(n, m) (Stirling2).
a(n) = (2^n - 64*4^n + 486*6^n - 1024*8^n + 625*10^n)/4!.
G.f.: 1/((1-2*x)*(1-4*x)*(1-6*x)*(1-8*x)*(1-10*x)).
E.g.f.: (d^5/dx^5)(((exp(2*x)-1)/2)^5)/5! = (exp(2*x) - 64*exp(4*x) + 486*exp(6*x) - 1024*exp(8*x) + 625*exp(10*x))/4!.

A075516 Sixth column of triangle A075498.

Original entry on oeis.org

1, 63, 2394, 71442, 1848987, 43615341, 964942308, 20385709344, 416206043253, 8280505692459, 161494678323342, 3101091077181006, 58823743379417199, 1104995938593100617, 20595841868175915096
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(3*(m+1)*x)/5!.

Crossrefs

Cf. A008277, A075498, A075513, A075515, A075906.

Formula

a(n) = A075498(n+6, 6) = (3^n)*S2(n+6, 6) with S2(n, m) := A008277(n, m) (Stirling2).
a(n) = Sum_{m=0..5} A075513(6, m)*((m+1)*3)^n/5!.
G.f.: 1/Product_{k=1..6} (1 - 3*k*x).
E.g.f.: (d^6/dx^6)(((exp(3*x)-1)/3)^6)/6! = (-exp(3*x) + 160*exp(6*x) - 2430*exp(9*x) + 10240*exp(12*x) - 15625*exp(15*x) + 7776*exp(18*x))/5!.

A075913 Fifth column of triangle A075500.

Original entry on oeis.org

1, 75, 3500, 131250, 4344375, 132890625, 3855156250, 107765625000, 2933008203125, 78271552734375, 2058270703125000, 53524929199218750, 1380066321044921875, 35349237725830078125, 900813505310058593750, 22863955398559570312500, 578500758117828369140625
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(5*(m+1)*x))/4!.

Links

Crossrefs

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

Programs

  • Mathematica
    Table[5^n*(1 - 2^(n+6) + 2*3^(n+5) - 4^(n+5) + 5^(n+4))/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)) + O(x^30)) \\ Colin Barker, Dec 12 2015

Formula

a(n) = A075500(n+5, 5) = (5^n)*S2(n+5, 5) with S2(n, m) = A008277(n, m) (Stirling2).
a(n) = Sum_{m=0..4}(A075513(5, m)*((m+1)*5)^n)/4!.
G.f.: 1/Product_{k=1..5}(1-5*k*x).
E.g.f.: (d^5/dx^5)((((exp(5*x)-1)/5)^5)/5!) = (exp(5*x) - 64*exp(10*x) + 486*exp(15*x) - 1024*exp(20*x) + 625*exp(25*x))/4!.
G.f.: 1 / ((1-5*x)*(1-10*x)*(1-15*x)*(1-20*x)*(1-25*x)). - Colin Barker, Dec 12 2015
Previous Showing 11-20 of 55 results. Next

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