A362381 -id:A362381 - cp's OEIS frontend

A362378 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..floor(n/3)} (k/6)^j * (j+1)^(n-2j-1) / (j! (n-3*j)!).

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 9, 1, 1, 1, 1, 4, 17, 41, 1, 1, 1, 1, 5, 25, 81, 191, 1, 1, 1, 1, 6, 33, 121, 441, 1191, 1, 1, 1, 1, 7, 41, 161, 751, 3641, 9353, 1, 1, 1, 1, 8, 49, 201, 1121, 7351, 33825, 77897, 1

Offset: 0

Seiichi Manyama, Apr 20 2023

			Square array begins:
  1,   1,   1,   1,    1,    1,    1, ...
  1,   1,   1,   1,    1,    1,    1, ...
  1,   1,   1,   1,    1,    1,    1, ...
  1,   2,   3,   4,    5,    6,    7, ...
  1,   9,  17,  25,   33,   41,   49, ...
  1,  41,  81, 121,  161,  201,  241, ...
  1, 191, 441, 751, 1121, 1551, 2041, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened
Eric Weisstein's World of Mathematics, Lambert W-Function.

Columns k=0..3 give A000012, A362381, A362390, A362391.

Cf. A362043, A362377.

T(n, k) = n! * sum(j=0, n\3, (k/6)^j*(j+1)^(n-2*j-1)/(j!*(n-3*j)!));

E.g.f. A_k(x) of column k satisfies A_k(x) = exp(x + k*x^3/6 * A_k(x)).
A_k(x) = exp(x - LambertW(-k*x^3/6 * exp(x))).
A_k(x) = -6 * LambertW(-k*x^3/6 * exp(x))/(k*x^3) for k > 0.

A362477 E.g.f. satisfies A(x) = exp(x + x^3/6 * A(x)^3).

Original entry on oeis.org

1, 1, 1, 2, 17, 161, 1351, 12391, 153385, 2388905, 40060781, 708351821, 13861042801, 305141790097, 7339275555067, 188198812659131, 5143808931521681, 150713978752271441, 4718460264313196665, 156524510548008965305, 5474266337362911068161

Offset: 0

Seiichi Manyama, Apr 21 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..408
Eric Weisstein's World of Mathematics, Lambert W-Function.

Column k=1 of A362490.

Cf. A362381.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x-lambertw(-x^3/2*exp(3*x))/3)))

E.g.f.: exp(x - LambertW(-x^3/2 * exp(3*x))/3) = ( -2 * LambertW(-x^3/2 * exp(3*x))/x^3 )^(1/3).

a(n) = n! * Sum_{k=0..floor(n/3)} (1/6)^k * (3*k+1)^(n-2*k-1) / (k! * (n-3*k)!).

A386531 E.g.f. A(x) satisfies A(x) = exp(x + x^3/6 * A''(x)).

Original entry on oeis.org

1, 1, 1, 2, 13, 181, 4551, 188021, 11924753, 1103029649, 142906232381, 25095114042461, 5813156139567261, 1736262706526700925, 655797361805578202939, 308047913827328021014851, 177358895717746915172030241, 123578165227603044619210348321

Offset: 0

Seiichi Manyama, Jul 24 2025

nonn

Cf. A385322, A386532.

Cf. A362381, A385066.

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=v[i]+sum(j=0, i-1, (1+j)*sum(k=1, 2, stirling(2, k, 1)*j^k)*binomial(i-1, j)*v[j+1]*v[i-j])/6); v;

a(0) = 1; a(n) = a(n-1) + (1/6) * Sum_{k=0..n-1} (-k + k^3) * binomial(n-1,k) * a(k) * a(n-1-k).

A362748 E.g.f. satisfies A(x) = exp(x^3/6 + x * A(x)).

Original entry on oeis.org

1, 1, 3, 17, 133, 1386, 18097, 284299, 5225985, 110097836, 2616190831, 69236871309, 2019833025157, 64403044165942, 2228441614038837, 83166830262851591, 3330183199746011713, 142418071427679810936, 6478769455582913796475

Offset: 0

Seiichi Manyama, May 02 2023

nonn

Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A349562, A362747.

Cf. A362381, A362691.

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x^3/6-lambertw(-x*exp(x^3/6)))))

E.g.f.: -LambertW(-x * exp(x^3/6)) / x = exp( x^3/6 - LambertW(-x*exp(x^3/6)) ).

a(n) = n! * Sum_{k=0..floor(n/3)} (n-3*k+1)^(n-2*k-1) / (6^k * k! * (n-3*k)!).

cp's OEIS Frontend

A362378 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..floor(n/3)} (k/6)^j * (j+1)^(n-2j-1) / (j! (n-3*j)!).

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A362477 E.g.f. satisfies A(x) = exp(x + x^3/6 * A(x)^3).

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A386531 E.g.f. A(x) satisfies A(x) = exp(x + x^3/6 * A''(x)).

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A362748 E.g.f. satisfies A(x) = exp(x^3/6 + x * A(x)).

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cp's OEIS Frontend

A362378 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..floor(n/3)} (k/6)^j * (j+1)^(n-2*j-1) / (j! * (n-3*j)!).

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A362477 E.g.f. satisfies A(x) = exp(x + x^3/6 * A(x)^3).

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A386531 E.g.f. A(x) satisfies A(x) = exp(x + x^3/6 * A''(x)).

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Author

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A362748 E.g.f. satisfies A(x) = exp(x^3/6 + x * A(x)).

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A362378 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..floor(n/3)} (k/6)^j * (j+1)^(n-2j-1) / (j! (n-3*j)!).