A067414 -id:A067414 - 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).

A067413 Sixth column of triangle A067410.

Original entry on oeis.org

1, 7, 84, 1008, 12096, 145152, 1741824, 20901888, 250822656, 3009871872, 36118462464, 433421549568, 5201058594816, 62412703137792, 748952437653504, 8987429251842048, 107849151022104576, 1294189812265254912

Offset: 0

Wolfdieter Lang, Jan 25 2002

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

Index entries for linear recurrences with constant coefficients, signature (12).

Cf. A067412 (fourth column), A067414 (seventh column), A001021 (powers of 12).

a(n)= A067410(n+5, 5). a(n)= 7*12^(n-1), n>=1, a(0)=1.

G.f.: (1-5*x)/(1-12*x).

A067415 Eighth column of triangle A067410.

Original entry on oeis.org

1, 9, 144, 2304, 36864, 589824, 9437184, 150994944, 2415919104, 38654705664, 618475290624, 9895604649984, 158329674399744, 2533274790395904, 40532396646334464, 648518346341351424

Offset: 0

Wolfdieter Lang, Jan 25 2002

Index entries for linear recurrences with constant coefficients, signature (16).

Cf. A067414 (seventh column), A067416 (ninth column), A001025 (powers of 16).

a(n)= A067410(n+7, 7). a(n)= 9*16^(n-1), n>=1, a(0)=1.

G.f.: (1-7*x)/(1-16*x).

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A067410 Triangle with columns built from certain power sequences.

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A067413 Sixth column of triangle A067410.

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A067415 Eighth column of triangle A067410.

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