A119336 -id:A119336 - cp's OEIS frontend

A119335 Number triangle T(n,k) = Sum_{j=0..n-k} C(k,3j)*C(n-k,3j).

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 5, 5, 1, 1, 1, 1, 1, 1, 11, 17, 11, 1, 1, 1, 1, 1, 1, 21, 41, 41, 21, 1, 1, 1, 1, 1, 1, 36, 81, 101, 81, 36, 1, 1, 1, 1, 1, 1, 57, 141, 201, 201, 141, 57, 1, 1, 1

Offset: 0

Paul Barry, May 14 2006

Row sums are A119336. Product of Pascal's triangle and A119337.

			Triangle begins
1;
1, 1;
1, 1, 1;
1, 1, 1,  1;
1, 1, 1,  1,  1;
1, 1, 1,  1,  1,   1;
1, 1, 1,  2,  1,   1,  1;
1, 1, 1,  5,  5,   1,  1,  1;
1, 1, 1, 11, 17,  11,  1,  1, 1;
1, 1, 1, 21, 41,  41, 21,  1, 1, 1;
1, 1, 1, 36, 81, 101, 81, 36, 1, 1, 1;

Seiichi Manyama, Rows n = 0..139, flattened

T(2n,n) gives A119363.

Cf. A119326.

T[n_, k_] := Sum[Binomial[k, 3j] Binomial[n-k, 3j], {j, 0, n-k}];
Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Sep 14 2023 *)

Column k has g.f. (x^k/(1-x)) * Sum_{j=0..k} C(k,3j)(x/(1-x))^(3j).

More terms from Seiichi Manyama, Mar 12 2019

A307078 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. ((1-x)^(k-2))/((1-x)^k-x^k).

Original entry on oeis.org

1, 1, 3, 1, 2, 7, 1, 2, 4, 15, 1, 2, 3, 8, 31, 1, 2, 3, 5, 16, 63, 1, 2, 3, 4, 10, 32, 127, 1, 2, 3, 4, 6, 21, 64, 255, 1, 2, 3, 4, 5, 12, 43, 128, 511, 1, 2, 3, 4, 5, 7, 28, 86, 256, 1023, 1, 2, 3, 4, 5, 6, 14, 64, 171, 512, 2047, 1, 2, 3, 4, 5, 6, 8, 36, 136, 341, 1024, 4095

Offset: 0

Seiichi Manyama, Mar 22 2019

			Square array begins:
     1,   1,   1,   1,  1,  1,  1,  1, 1, ...
     3,   2,   2,   2,  2,  2,  2,  2, 2, ...
     7,   4,   3,   3,  3,  3,  3,  3, 3, ...
    15,   8,   5,   4,  4,  4,  4,  4, 4, ...
    31,  16,  10,   6,  5,  5,  5,  5, 5, ...
    63,  32,  21,  12,  7,  6,  6,  6, 6, ...
   127,  64,  43,  28, 14,  8,  7,  7, 7, ...
   255, 128,  86,  64, 36, 16,  9,  8, 8, ...
   511, 256, 171, 136, 93, 45, 18, 10, 9, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns 1-6 give A126646, A000079, A024494(n+1), A038504(n+1), A133476(n+1), A119336.

Cf. A306846, A306915, A307079.

T[n_, k_] := Sum[Binomial[n+1, k*j+1], {j, 0, Floor[n/k]}]; Table[T[n-k, k], {n, 0, 12}, {k, n, 1, -1}] // Flatten (* Amiram Eldar, May 20 2021 *)

A(n,k) = Sum_{j=0..floor(n/k)} binomial(n+1,k*j+1).

A(n,2*k) = Sum_{i=0..n} Sum_{j=0..n-i} binomial(i,k*j) * binomial(n-i,k*j).

A307089 Expansion of (1 - x)^4/((1 - x)^6 + x^6).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 6, 0, -27, -110, -319, -780, -1702, -3404, -6315, -10864, -17051, -23238, -23238, 0, 87021, 325358, 890077, 2107560, 4542526, 9085052, 16950573, 29354524, 46296905, 63239286, 63239286, 0, -236031147, -880918070, -2406788599, -5694626340

Offset: 0

Seiichi Manyama, Mar 24 2019

Seiichi Manyama, Table of n, a(n) for n = 0..3000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-2).

Column 6 of A307079.
Row sums of triangle A307156.
Cf. A119336.

a[n_] := Sum[(-1)^k * Binomial[n+1, 6*k+1], {k, 0, Floor[n/6]}]; Array[a, 36, 0] (* Amiram Eldar, May 14 2021 *)

{a(n) = sum(k=0, n\6, (-1)^k*binomial(n+1,6*k+1))}

N=66; x='x+O('x^N); Vec((1-x)^4/((1-x)^6+x^6))

a(n) = Sum_{k=0..floor(n/6)} (-1)^k*binomial(n+1,6*k+1).
a(n) = Sum_{i=0..n} Sum_{j=0..n-i} (-1)^j * binomial(i,3*j) * binomial(n-i,3*j).
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - 2*a(n-6) for n > 5.

cp's OEIS Frontend

A119335 Number triangle T(n,k) = Sum_{j=0..n-k} C(k,3j)*C(n-k,3j).

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A307078 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. ((1-x)^(k-2))/((1-x)^k-x^k).

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A307089 Expansion of (1 - x)^4/((1 - x)^6 + x^6).

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