A067413 -id:A067413 - cp's OEIS frontend

A067410 Triangle with columns built from certain power sequences.

1, 2, 1, 4, 3, 1, 8, 12, 4, 1, 16, 48, 24, 5, 1, 32, 192, 144, 40, 6, 1, 64, 768, 864, 320, 60, 7, 1, 128, 3072, 5184, 2560, 600, 84, 8, 1, 256, 12288, 31104, 20480, 6000, 1008, 112, 9, 1, 512, 49152, 186624, 163840, 60000, 12096, 1568, 144, 10, 1

Offset: 0

Wolfdieter Lang, Jan 25 2002

			Triangle starts:
  1;
  2,  1;
  4,  3, 1;
  8, 12, 4, 1;
  ...

Indranil Ghosh, Rows 0..125, flattened

Cf. A009998 (triangle built from powers of (m+1)), A067402.

Columns m=0..8 are A000079, A002001, A067411, A067412, A090019, A067413, A067414, A067415, A067416.

A[n_,m_]:=If[n==m,1,(m+2)(2(m+1))^(n-m-1)]; Flatten[Table[A[n,m],{n,0,9},{m,0,n}]] (* Stefano Spezia, Sep 30 2022 *)

a(n, m) = 1 if n = m; a(n, m) = (m+2)*(2*(m+1))^(n-m-1) if n > m >= 0.

G.f. for column m: (x^m)*(1-m*x)/(1-2*(m+1)*x).

A067412 Fourth column of triangle A067410.

Original entry on oeis.org

1, 5, 40, 320, 2560, 20480, 163840, 1310720, 10485760, 83886080, 671088640, 5368709120, 42949672960, 343597383680, 2748779069440, 21990232555520, 175921860444160, 1407374883553280, 11258999068426240

Offset: 0

Wolfdieter Lang, Jan 25 2002

The fifth column gives [1,6,60,600,6000,60000,...].

a(n+1) = A157176(A016957(n)). [From Reinhard Zumkeller, Feb 24 2009]

Index entries for linear recurrences with constant coefficients, signature (8).

Cf. A067411 (third column), A067413 (sixth column), A001018 (powers of 8).

Join[{1},NestList[8#&,5,20]] (* or *) CoefficientList[Series[ (1-3x)/ (1-8x),{x,0,20}],x] (* Harvey P. Dale, May 14 2011 *)

a(n)= A067410(n+3, 3). a(n)= 5*8^(n-1), n>=1, a(0)=1.
G.f.: (1-3*x)/(1-8*x).
E.g.f.: (5*exp(8*x)+3)/8 = exp(4*x)*(cosh(4*x)+sinh(4*x)/4) - Paul Barry, Nov 20 2003

A067414 Seventh column of triangle A067410.

Original entry on oeis.org

1, 8, 112, 1568, 21952, 307328, 4302592, 60236288, 843308032, 11806312448, 165288374272, 2314037239808, 32396521357312, 453551299002368, 6349718186033152, 88896054604464128, 1244544764462497792

Offset: 0

Wolfdieter Lang, Jan 25 2002

Index entries for linear recurrences with constant coefficients, signature (14).

Cf. A067413 (sixth column), A067415 (eighth column), A001023 (powers of 14).

a(n) = A067410(n+6, 6).
a(n) = 8*14^(n-1), n>=1, a(0)=1.
G.f.: (1-6*x)/(1-14*x).

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A067410 Triangle with columns built from certain power sequences.

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A067412 Fourth column of triangle A067410.

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A067414 Seventh column of triangle A067410.

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