A371675 -id:A371675 - cp's OEIS frontend

A378238 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(3n+r+k,n)/(3n+r+k) for k > 0.

1, 1, 0, 1, 2, 0, 1, 4, 14, 0, 1, 6, 32, 134, 0, 1, 8, 54, 324, 1482, 0, 1, 10, 80, 578, 3696, 17818, 0, 1, 12, 110, 904, 6810, 45316, 226214, 0, 1, 14, 144, 1310, 11008, 85278, 583152, 2984206, 0, 1, 16, 182, 1804, 16490, 140936, 1113854, 7769348, 40503890, 0

Offset: 0

Seiichi Manyama, Nov 20 2024

			Square array begins:
  1,      1,      1,       1,       1,       1,       1, ...
  0,      2,      4,       6,       8,      10,      12, ...
  0,     14,     32,      54,      80,     110,     144, ...
  0,    134,    324,     578,     904,    1310,    1804, ...
  0,   1482,   3696,    6810,   11008,   16490,   23472, ...
  0,  17818,  45316,   85278,  140936,  216002,  314700, ...
  0, 226214, 583152, 1113854, 1870352, 2914790, 4320608, ...

Columns k=0..3 give A000007, A144097, A371675, A365843.
T(n,n) gives 1/4 * A370102(n) for n > 0.
Cf. A033877, A071949, A378236, A378237, A378239, A378240.

T(n, k, t=3, u=1) = 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)^(3/k) * (1 + A_k(x)^(1/k)) )^k for k > 0.
G.f. of column k: B(x)^k where B(x) is the g.f. of A144097.
B(x)^k = B(x)^(k-1) + x * B(x)^(k+2) + x * B(x)^(k+3). So T(n,k) = T(n,k-1) + T(n-1,k+2) + T(n-1,k+3) for n > 0.

A371676 G.f. satisfies A(x) = 1 + x * A(x)^2 * (1 + A(x)^(1/2))^2.

Original entry on oeis.org

1, 4, 40, 524, 7824, 126228, 2143544, 37750812, 683194912, 12628104740, 237388091208, 4524456276524, 87228274533040, 1698091537435444, 33332913873239640, 659038408936005692, 13112372856351746112, 262338658739430857796, 5274545338183090647656

Offset: 0

Seiichi Manyama, Apr 02 2024

nonn

Jun Yan, Lattice paths enumerations weighted by ascent lengths, arXiv:2501.01152 [math.CO], 2025. See p. 7.

Cf. A006319, A032349, A371675. A371677, A371678.

Cf. A260332.

a(n, r=2, t=4, u=1) = r*sum(k=0, n, binomial(n, k)*binomial(t*n+u*k+r, n)/(t*n+u*k+r));

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

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

A371678 G.f. satisfies A(x) = 1 + x * A(x)^3 * (1 + A(x)^(1/2))^2.

Original entry on oeis.org

1, 4, 56, 1068, 23504, 561972, 14183880, 371911132, 10031990560, 276589937892, 7759696110808, 220805824681740, 6357540660485616, 184876232243020564, 5422016433851400552, 160187931368799105468, 4763038761416835095616, 142426926824923660491716

Offset: 0

Seiichi Manyama, Apr 02 2024

nonn

Jun Yan, Lattice paths enumerations weighted by ascent lengths, arXiv:2501.01152 [math.CO], 2025. See p. 7.

Cf. A006319, A032349, A371675. A371676, A371677.

Cf. A371700.

a(n, r=2, t=6, u=1) = r*sum(k=0, n, binomial(n, k)*binomial(t*n+u*k+r, n)/(t*n+u*k+r));

G.f. satisfies A(x) = ( 1 + x * A(x)^3 * (1 + A(x)^(1/2)) )^2.
G.f.: B(x)^2 where B(x) is the g.f. of A371700.
a(n) = 2 * Sum_{k=0..n} binomial(n,k) * binomial(6*n+k+2,n)/(6*n+k+2).

A371677 G.f. satisfies A(x) = 1 + x * A(x)^(5/2) * (1 + A(x)^(1/2))^2.

Original entry on oeis.org

1, 4, 48, 772, 14256, 285380, 6023552, 131991940, 2974096544, 68475379204, 1603913377040, 38099316926340, 915619571011024, 22222175033464260, 543894269096547296, 13409307961403740420, 332707806061304185408, 8301493488646359256580

Offset: 0

Seiichi Manyama, Apr 02 2024

nonn

Cf. A006319, A032349, A371675. A371676, A371678.

Cf. A363006.

a(n, r=2, t=5, u=1) = r*sum(k=0, n, binomial(n, k)*binomial(t*n+u*k+r, n)/(t*n+u*k+r));

G.f. satisfies A(x) = ( 1 + x * A(x)^(5/2) * (1 + A(x)^(1/2)) )^2.
G.f.: B(x)^2 where B(x) is the g.f. of A363006.
a(n) = 2 * Sum_{k=0..n} binomial(n,k) * binomial(5*n+k+2,n)/(5*n+k+2).

A379282 G.f. A(x) satisfies A(x) = 1/( (1 - xA(x)^2) (1 - x*A(x)) )^2.

Original entry on oeis.org

1, 4, 34, 376, 4743, 64710, 929906, 13865206, 212509079, 3327383632, 52994140217, 855842582128, 13982509284464, 230686414552016, 3837897905208588, 64314848237403878, 1084624929809399857, 18393856772155371200, 313487249756740510907, 5366521088581773011788

Offset: 0

Seiichi Manyama, Dec 19 2024

nonn

Cf. A371675, A379251, A379279.
Cf. A379281, A379283.
Cf. A379284.

a(n) = 2*sum(k=0, n, binomial(2*n+3*k+2, k)*binomial(3*n+k+1, n-k)/(2*n+3*k+2));

G.f.: B(x)^2 where B(x) is the g.f. of A379284.

a(n) = 2 * Sum_{k=0..n} binomial(2*n+3*k+2,k) * binomial(3*n+k+1,n-k)/(2*n+3*k+2).

A379244 G.f. A(x) satisfies A(x) = ( (1 + xA(x)^3)/(1 - xA(x)) )^2.

Original entry on oeis.org

1, 4, 40, 540, 8400, 141876, 2528760, 46815116, 891483808, 17350187364, 343578992328, 6900588813564, 140230648164720, 2878066866407316, 59571280942854808, 1242093725341221996, 26064579113472078144, 550041399791036747460, 11665771061882347813224, 248527169321049466503132

Offset: 0

Seiichi Manyama, Dec 18 2024

nonn

Cf. A032349, A371675.

Cf. A364167, A379174.

a(n) = sum(k=0, n, binomial(2*n+4*k+2, k)*binomial(3*n+3*k+1, n-k)/(n+2*k+1));

G.f.: B(x)^2 where B(x) is the g.f. of A364167.

a(n) = Sum_{k=0..n} binomial(2*n+4*k+2,k) * binomial(3*n+3*k+1,n-k)/(n+2*k+1).

A379279 G.f. A(x) satisfies A(x) = ( (1 + xA(x)^2) (1 + x*A(x)) )^2.

Original entry on oeis.org

1, 4, 30, 288, 3125, 36490, 447478, 5683186, 74105002, 986302778, 13344661479, 182998935930, 2537838036761, 35530970858236, 501523116910044, 7129275916213606, 101973703002773268, 1466574750062589956, 21194869324964207133, 307642575576365729486, 4482940969372057898247

Offset: 0

Seiichi Manyama, Dec 19 2024

nonn

Cf. A371675, A379251, A379282.
Cf. A073155, A379280.
Cf. A215715.

a(n) = sum(k=0, n, binomial(2*n+2*k+2, k)*binomial(2*n+2*k+2, n-k)/(n+k+1));

G.f.: B(x)^2 where B(x) is the g.f. of A215715.

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

A371679 G.f. satisfies A(x) = ( 1 + x * A(x)^(3/2) * (1 + A(x)) )^2.

Original entry on oeis.org

1, 4, 36, 424, 5696, 82720, 1264816, 20060512, 326990528, 5444291968, 92193926528, 1582961928448, 27493991536384, 482203526685696, 8527881803412224, 151909590806619648, 2723133151505640448, 49087220319316809728, 889230405958421051392

Offset: 0

Seiichi Manyama, Apr 02 2024

nonn

Cf. A363311, A371675.

a(n, r=2, t=3, 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 A363311.

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

cp's OEIS Frontend

A378238 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(3*n+r+k,n)/(3*n+r+k) for k > 0.

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A371676 G.f. satisfies A(x) = 1 + x * A(x)^2 * (1 + A(x)^(1/2))^2.

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A371678 G.f. satisfies A(x) = 1 + x * A(x)^3 * (1 + A(x)^(1/2))^2.

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A371677 G.f. satisfies A(x) = 1 + x * A(x)^(5/2) * (1 + A(x)^(1/2))^2.

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A379282 G.f. A(x) satisfies A(x) = 1/( (1 - x*A(x)^2) * (1 - x*A(x)) )^2.

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A379244 G.f. A(x) satisfies A(x) = ( (1 + x*A(x)^3)/(1 - x*A(x)) )^2.

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A379279 G.f. A(x) satisfies A(x) = ( (1 + x*A(x)^2) * (1 + x*A(x)) )^2.

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A371679 G.f. satisfies A(x) = ( 1 + x * A(x)^(3/2) * (1 + A(x)) )^2.

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Crossrefs

Programs

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Formula

A378238 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(3n+r+k,n)/(3n+r+k) for k > 0.

A379282 G.f. A(x) satisfies A(x) = 1/( (1 - xA(x)^2) (1 - x*A(x)) )^2.

A379244 G.f. A(x) satisfies A(x) = ( (1 + xA(x)^3)/(1 - xA(x)) )^2.

A379279 G.f. A(x) satisfies A(x) = ( (1 + xA(x)^2) (1 + x*A(x)) )^2.