A067419 -id:A067419 - cp's OEIS frontend

A068180 (Product_{i=1..4} (x+i)) / (Product_{i=1..4} (x-i)) = Sum_{n>=1} a(n)/A067419(n)*x^n.

1, 25, 625, 11095, 164125, 2201575, 28021525, 346791295, 4228592125, 51161968375, 616523997925, 7414045240495, 89064205082125, 1069348964379175, 12835676881182325, 154049132081273695

Offset: 1

Benoit Cloitre, Mar 12 2002

Vincenzo Librandi, Table of n, a(n) for n = 1..900
Index entries for linear recurrences with constant coefficients, signature (25,-210,720,-864).

Cf. A067419.

LinearRecurrence[{25,-210,720,-864},{1,25,625,11095,164125},30] (* Harvey P. Dale, Oct 28 2015 *)

Lim_{n->infinity} a(n)/A067419(n) = 20.
For n > 1, a(n) = (5/6)*12^n - (15/2)*6^n + (35/2)*4^n - (35/3)*3^n. - Ralf Stephan, May 08 2004
G.f.: x*(864*x^4 + 210*x^2 + 1) / ((3*x-1)*(4*x-1)*(6*x-1)*(12*x-1)). - Colin Barker, Jun 17 2013

A055372 Invert transform of Pascal's triangle A007318.

Original entry on oeis.org

1, 1, 1, 2, 4, 2, 4, 12, 12, 4, 8, 32, 48, 32, 8, 16, 80, 160, 160, 80, 16, 32, 192, 480, 640, 480, 192, 32, 64, 448, 1344, 2240, 2240, 1344, 448, 64, 128, 1024, 3584, 7168, 8960, 7168, 3584, 1024, 128, 256, 2304, 9216, 21504, 32256, 32256, 21504, 9216, 2304, 256

Offset: 0

Christian G. Bower, May 16 2000

Triangle T(n,k), 0 <= k <= n, read by rows, given by [1, 1, 0, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 1, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Aug 10 2005

T(n,k) is the number of nonempty bit strings with n bits and exactly k 1's over all strings in the sequence. For example, T(2,1)=4 because we have {(01)},{(10)},{(0),(1)},{(1),(0)}. - Geoffrey Critzer, Apr 06 2013

			Triangle begins:
  1;
  1,  1;
  2,  4,  2;
  4, 12, 12,  4;
  8, 32, 48, 32,  8;
  ...

N. J. A. Sloane, Transforms
Index entries for triangles and arrays related to Pascal's triangle

Row sums give A081294. Cf. A000079, A007318, A055373, A055374.
Cf. A134309.
T(2n,n) gives A098402.

nn=10;f[list_]:=Select[list,#>0&];a=(x+y x)/(1-(x+y x));Map[f,CoefficientList[Series[1/(1-a),{x,0,nn}],{x,y}]]//Grid  (* Geoffrey Critzer, Apr 06 2013 *)

a(n,k) = 2^(n-1)*C(n, k), for n>0.
G.f.: A(x, y)=(1-x-xy)/(1-2x-2xy).
As an infinite lower triangular matrix, equals A134309 * A007318. - Gary W. Adamson, Oct 19 2007
Sum_{k=0..n} T(n,k)*x^k = A000007(n), A011782(n), A081294(n), A081341(n), A092811(n), A093143(n), A067419(n) for x = -1, 0, 1, 2, 3, 4, 5 respectively. - Philippe Deléham, Feb 05 2012

A134309 Triangle read by rows, where row n consists of n zeros followed by 2^(n-1).

Original entry on oeis.org

1, 0, 1, 0, 0, 2, 0, 0, 0, 4, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 16, 0, 0, 0, 0, 0, 0, 32, 0, 0, 0, 0, 0, 0, 0, 64, 0, 0, 0, 0, 0, 0, 0, 0, 128, 0, 0, 0, 0, 0, 0, 0, 0, 0, 256, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 512, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1024, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2048, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Gary W. Adamson, Oct 19 2007

As infinite lower triangular matrices, binomial transform of A134309 = A082137. A134309 * A007318 = A055372. A134309 * [1,2,3,...] = A057711: (1, 2, 6, 16, 40, 96, 224,...).

Triangle read by rows given by [0,0,0,0,0,0,0,0,...] DELTA [1,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 20 2007

			Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
  1;
  0, 1;
  0, 0, 2;
  0, 0, 0, 4;
  0, 0, 0, 0, 8;
  0, 0, 0, 0, 0, 16;
  ...

Cf. A011782 (diagonal elements: 1 followed by 1, 2, 4, 8, ... = A000079: 2^n).

Cf. A055372, A057711, A082137.

Join[{1},Flatten[Table[Join[{PadRight[{},n],2^(n-1)}],{n,20}]]] (* Harvey P. Dale, Jan 04 2024 *)

A134309(r,c)=if(r==c,2^max(r-1,0),0) \\ M. F. Hasler, Mar 29 2022

Triangle, T(0,0) = 1, then for n > 0, n zeros followed by 2^(n-1). Infinite lower triangular matrix with (1, 1, 2, 4, 8, 16, ...) in the main diagonal and the rest zeros.
G.f.: (1 - y*x)/(1 - 2*y*x). - Philippe Deléham, Feb 04 2012
Sum_{k=0..n} T(n,k)*x^k = A000007(n), A011782(n), A081294(n), A081341(n), A092811(n), A093143(n), A067419(n) for x = 0, 1, 2, 3, 4, 5, 6 respectively. - Philippe Deléham, Feb 04 2012
Diagonal is A011782, other elements are 0. - M. F. Hasler, Mar 29 2022

A067417 Triangle with columns built from certain power sequences.

Original entry on oeis.org

1, 3, 1, 9, 4, 1, 27, 24, 5, 1, 81, 144, 45, 6, 1, 243, 864, 405, 72, 7, 1, 729, 5184, 3645, 864, 105, 8, 1, 2187, 31104, 32805, 10368, 1575, 144, 9, 1, 6561, 186624, 295245, 124416, 23625, 2592, 189, 10, 1, 19683, 1119744, 2657205, 1492992, 354375, 46656, 3969, 240, 11, 1

Offset: 0

Wolfdieter Lang, Jan 25 2002

			Triangle starts:
   1;
   3,  1;
   9,  4, 1;
  27, 24, 5, 1;
  ...

Indranil Ghosh, Rows 0..125, flattened

Cf. A009998 (triangle built from powers of (m+1)), A067402, A067410.

Columns m=0..8 are A000244, A067411, A067403, A067419, A067420, A067421, A067422, A067423, A067424.

A[n_,m_]:=If[n==m,1,(m+3)(3(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+3)*(3*(m+1))^(n-m-1) if n > m >= 0.

G.f. for column m: (x^m)*(1-2*m*x)/(1-3*(m+1)*x).

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).

A067420 Fifth column of triangle A067417.

Original entry on oeis.org

1, 7, 105, 1575, 23625, 354375, 5315625, 79734375, 1196015625, 17940234375, 269103515625, 4036552734375, 60548291015625, 908224365234375, 13623365478515625, 204350482177734375, 3065257232666015625

Offset: 0

Wolfdieter Lang, Jan 25 2002

Vincenzo Librandi, Table of n, a(n) for n = 0..800
Index entries for linear recurrences with constant coefficients, signature (15).

Cf. A067419 (fourth column), A067421 (sixth column), A001024 (powers of 15).

[Ceiling(7*(3*5)^(n-1)): n in [0..20]]; // Vincenzo Librandi, Oct 02 2011

a(n) = A067417(n+4, 4).
a(n) = 7*(3*5)^(n-1), n >= 1, a(0)=1.
G.f.: (1-8*x)/(1-15*x).

cp's OEIS Frontend

A068180 (Product_{i=1..4} (x+i)) / (Product_{i=1..4} (x-i)) = Sum_{n>=1} a(n)/A067419(n)*x^n.

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A055372 Invert transform of Pascal's triangle A007318.

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A134309 Triangle read by rows, where row n consists of n zeros followed by 2^(n-1).

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PARI

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A067417 Triangle with columns built from certain power sequences.

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A067425 Triangle with columns built from certain power sequences.

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A067420 Fifth column of triangle A067417.

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