A371693 -id:A371693 - cp's OEIS frontend

A378239 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,0) = 0^n and T(n,k) = k * Sum_{r=0..n} binomial(n,r) * binomial(2n+2r+k,n)/(2n+2r+k) for k > 0.

1, 1, 0, 1, 2, 0, 1, 4, 12, 0, 1, 6, 28, 100, 0, 1, 8, 48, 248, 968, 0, 1, 10, 72, 452, 2480, 10208, 0, 1, 12, 100, 720, 4680, 26688, 113792, 0, 1, 14, 132, 1060, 7728, 51504, 301648, 1318832, 0, 1, 16, 168, 1480, 11800, 87104, 591312, 3531424, 15732064, 0

Offset: 0

Seiichi Manyama, Nov 20 2024 based on suggestions from Mikhail Kurkov

			Square array begins:
  1,      1,      1,      1,       1,       1,       1, ...
  0,      2,      4,      6,       8,      10,      12, ...
  0,     12,     28,     48,      72,     100,     132, ...
  0,    100,    248,    452,     720,    1060,    1480, ...
  0,    968,   2480,   4680,    7728,   11800,   17088, ...
  0,  10208,  26688,  51504,   87104,  136352,  202560, ...
  0, 113792, 301648, 591312, 1017184, 1621280, 2454256, ...

Columns k=0..4 give A000007, A219534, A371693, A378155, A378156.

Cf. A033877, A071949, A378236, A378237, A378238, A378240.

T(n, k, t=2, u=2) = if(k==0, 0^n, k*sum(r=0, n, binomial(n, r)*binomial(t*n+u*r+k, n)/(t*n+u*r+k)));
matrix(7, 7, n, k, T(n-1, k-1))

G.f. A_k(x) of column k satisfies A_k(x) = ( 1 + x * A_k(x)^(2/k) * (1 + A_k(x)^(2/k)) )^k for k > 0.
G.f. of column k: B(x)^k where B(x) is the g.f. of A219534.
B(x)^k = B(x)^(k-1) + x * B(x)^(k+1) + x * B(x)^(k+3). So T(n,k) = T(n,k-1) + T(n-1,k+1) + T(n-1,k+3) for n > 0.

A371681 G.f. satisfies A(x) = ( 1 + x * A(x) * (1 + A(x)) )^3.

Original entry on oeis.org

1, 6, 66, 926, 14706, 251622, 4524786, 84310014, 1613384994, 31521329670, 626151135330, 12608193099294, 256769542135314, 5279533270393446, 109449833201392530, 2285215031994672894, 48011502768234360642, 1014265693597636966662

Offset: 0

Seiichi Manyama, Apr 02 2024

nonn

Cf. A006318, A371693.

Cf. A364167.

a(n, r=3, t=3, u=3) = r*sum(k=0, n, binomial(n, k)*binomial(t*n+u*k+r, n)/(t*n+u*k+r));

G.f.: B(x)^3 where B(x) is the g.f. of A364167.

a(n) = Sum_{k=0..n} binomial(n,k) * binomial(3*n+3*k+3,n)/(n+k+1).

A378155 G.f. A(x) satisfies A(x) = ( 1 + x * A(x)^(2/3) * (1 + A(x)^(2/3)) )^3.

Original entry on oeis.org

1, 6, 48, 452, 4680, 51504, 591312, 7002864, 84926304, 1049402944, 13165069824, 167239042176, 2146912312064, 27808372643328, 362981425115904, 4769884412086016, 63050983340533248, 837805424714425344, 11184489029495865344, 149935005483457542144, 2017560365768892739584

Offset: 0

Seiichi Manyama, Nov 18 2024

nonn

Cf. A219534, A371693, A378156.

a(n, r=3, t=2, u=2) = r*sum(k=0, n, binomial(n, k)*binomial(t*n+u*k+r, n)/(t*n+u*k+r));

G.f.: B(x)^3 where B(x) is the g.f. of A219534.

a(n) = 3 * Sum_{k=0..n} binomial(n,k) * binomial(2*n+2*k+3,n)/(2*n+2*k+3).

A378156 G.f. A(x) satisfies A(x) = ( 1 + x * A(x)^(1/2) * (1 + A(x)^(1/2)) )^4.

Original entry on oeis.org

1, 8, 72, 720, 7728, 87104, 1017184, 12200640, 149429504, 1861059328, 23498407680, 300110580224, 3870135336192, 50323754919936, 659085377250816, 8686436702866432, 115120162870534144, 1533214282017157120, 20510220228874399744, 275462154992599851008, 3712900128220039372800

Offset: 0

Seiichi Manyama, Nov 18 2024

nonn

Cf. A219534, A371693, A378155.

a(n, r=4, t=2, u=2) = r*sum(k=0, n, binomial(n, k)*binomial(t*n+u*k+r, n)/(t*n+u*k+r));

G.f.: B(x)^4 where B(x) is the g.f. of A219534.

a(n) = 2 * Sum_{k=0..n} binomial(n,k) * binomial(2*n+2*k+4,n)/(n+k+2).

A371680 G.f. satisfies A(x) = ( 1 + x * A(x)^2 * (1 + A(x)) )^2.

Original entry on oeis.org

1, 4, 44, 648, 10960, 200992, 3886928, 78043488, 1611405504, 33998715264, 729793915264, 15886841223936, 349900041893376, 7782694227059712, 174573007616191744, 3944500600180286976, 89696369377912622080, 2051147782339517224960

Offset: 0

Seiichi Manyama, Apr 02 2024

nonn

Cf. A363380, A371693.

a(n, r=2, t=4, u=2) = r*sum(k=0, n, binomial(n, k)*binomial(t*n+u*k+r, n)/(t*n+u*k+r));

G.f.: B(x)^2 where B(x) is the g.f. of A363380.

a(n) = Sum_{k=0..n} binomial(n,k) * binomial(4*n+2*k+2,n)/(2*n+k+1).

cp's OEIS Frontend

A378239 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,0) = 0^n and T(n,k) = k * Sum_{r=0..n} binomial(n,r) * binomial(2*n+2*r+k,n)/(2*n+2*r+k) for k > 0.

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A371681 G.f. satisfies A(x) = ( 1 + x * A(x) * (1 + A(x)) )^3.

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A378155 G.f. A(x) satisfies A(x) = ( 1 + x * A(x)^(2/3) * (1 + A(x)^(2/3)) )^3.

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A378156 G.f. A(x) satisfies A(x) = ( 1 + x * A(x)^(1/2) * (1 + A(x)^(1/2)) )^4.

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A371680 G.f. satisfies A(x) = ( 1 + x * A(x)^2 * (1 + A(x)) )^2.

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Formula

A378239 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,0) = 0^n and T(n,k) = k * Sum_{r=0..n} binomial(n,r) * binomial(2n+2r+k,n)/(2n+2r+k) for k > 0.