A067404 -id:A067404 - cp's OEIS frontend

A067403 Third column of triangle A067402.

1, 5, 45, 405, 3645, 32805, 295245, 2657205, 23914845, 215233605, 1937102445, 17433922005, 156905298045, 1412147682405, 12709329141645, 114383962274805, 1029455660473245, 9265100944259205, 83385908498332845, 750473176484995605, 6754258588364960445, 60788327295284644005

Offset: 0

Wolfdieter Lang, Jan 25 2002

Index entries for linear recurrences with constant coefficients, signature (9).

Cf. A002001 (second column), A067404 (fourth column), A001019 (powers of 9).

Cf. A067402.

A067403:=n->5*9^(n-1): 1,seq(A067403(n), n=1..30); # Wesley Ivan Hurt, Apr 09 2017

Join[{1},NestList[9#&,5,30]] (* or *) CoefficientList[Series[ (1-4x)/ (1-9x),{x,0,30}],x] (* Harvey P. Dale, Apr 26 2011 *)

Vec((1-4*x)/(1-9*x) + O(x^30)) \\ Michel Marcus, Apr 09 2017

a(n) = A067402(n+2, 2).
a(n) = 5*9^(n-1) for n>=1, a(0) = 1.
G.f.: (1-4*x)/(1-9*x).
E.g.f.: (4 + 5*exp(9*x))/9. - Stefano Spezia, Sep 30 2022

A067425 Triangle with columns built from certain power sequences.

Original entry on oeis.org

1, 4, 1, 16, 5, 1, 64, 40, 6, 1, 256, 320, 72, 7, 1, 1024, 2560, 864, 112, 8, 1, 4096, 20480, 10368, 1792, 160, 9, 1, 16384, 163840, 124416, 28672, 3200, 216, 10, 1, 65536, 1310720, 1492992, 458752, 64000

Offset: 0

Wolfdieter Lang, Jan 25 2002

The fifth column (m=4) gives [1, 8, 160, 3200, 64000, 1280000, 25600000, ...].

			Triangle starts:
   1;
   4,  1;
  16,  5,  1;
  64, 40,  6,  1;
  ...

Indranil Ghosh, Rows 0..125, flattened

Cf. A009998, A067410, A067417.
Columns 0..3 are A000302 (powers of 4), A067412, A067419, A067404.
Columns 5..8 are A067426, A067427, A067428, A067429.

A067425[n_, m_] := If[n == m, 1, (m + 4)*(4*(m + 1))^(n - m - 1)];
Table[A067425[n, m], {n, 0, 10}, {m, 0, n}] (* Paolo Xausa, Oct 16 2024 *)

T(n,m) = 1 if n = m; T(n,m) = (m+4)*(4*(m+1))^(n-m-1) if n > m >= 0, else 0.

G.f. for column m: (x^m)*(1-3*m*x)/(1-4*(m+1)*x).

A067426 Sixth column of triangle A067425.

Original entry on oeis.org

1, 9, 216, 5184, 124416, 2985984, 71663616, 1719926784, 41278242816, 990677827584, 23776267862016, 570630428688384, 13695130288521216, 328683126924509184, 7888395046188220416, 189321481108517289984

Offset: 0

Wolfdieter Lang, Jan 25 2002

The fifth column of A067425 gives [1, 8, 160, 3200, 64000, 1280000, 25600000, ...].

Vincenzo Librandi, Table of n, a(n) for n = 0..700
Index entries for linear recurrences with constant coefficients, signature (24).

Cf. A009968 (powers of 24), A067404 (fourth column), A067425, A067427 (seventh column).

[Ceiling(9*(4*6)^(n-1)): n in [0..20]]; // Vincenzo Librandi, Oct 02 2011

Table[Ceiling[9(24)^(n-1)],{n,0,15}] (* Paul F. Marrero Romero, Oct 11 2024 *)

a(n) = A067425(n+5, 5).
a(n) = 9*(4*6)^(n-1), n >= 1, a(0)=1.
G.f.: (1-15*x)/(1-24*x).
E.g.f.: (3*exp(24*x) + 5)/8. - Stefano Spezia, Oct 15 2024

A067405 Fifth column of triangle A067402.

Original entry on oeis.org

1, 9, 225, 5625, 140625, 3515625, 87890625, 2197265625, 54931640625, 1373291015625, 34332275390625, 858306884765625, 21457672119140625, 536441802978515625, 13411045074462890625, 335276126861572265625, 8381903171539306640625, 209547579288482666015625, 5238689482212066650390625

Offset: 0

Wolfdieter Lang, Jan 25 2002

Index entries for linear recurrences with constant coefficients, signature (25).

Cf. A067404 (fourth column), A067406 (sixth column), A009969 (powers of 25).

Cf. A067402.

Join[{1},NestList[25#&,9,20]] (* Harvey P. Dale, Mar 12 2023 *)

a(n) = A067402(n+4, 4).
a(n) = 9*25^(n-1), n>=1, a(0) = 1.
G.f.: (1-16*x)/(1-25*x).
E.g.f.: (16 + 9*exp(25*x))/25. - Stefano Spezia, Sep 30 2022

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A067403 Third column of triangle A067402.

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A067425 Triangle with columns built from certain power sequences.

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A067426 Sixth column of triangle A067425.

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A067405 Fifth column of triangle A067402.

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