A362378 -id:A362378 - cp's OEIS frontend

A362377 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..floor(n/2)} (k/2)^j * (j+1)^(n-j-1) / (j! * (n-2*j)!).

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 7, 1, 1, 1, 4, 13, 34, 1, 1, 1, 5, 19, 85, 216, 1, 1, 1, 6, 25, 154, 701, 1696, 1, 1, 1, 7, 31, 241, 1456, 7261, 15898, 1, 1, 1, 8, 37, 346, 2481, 18136, 89125, 173468, 1, 1, 1, 9, 43, 469, 3776, 35761, 260002, 1277865, 2161036, 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,    2,    3,     4,     5,     6,     7, ...
  1,    7,   13,    19,    25,    31,    37, ...
  1,   34,   85,   154,   241,   346,   469, ...
  1,  216,  701,  1456,  2481,  3776,  5341, ...
  1, 1696, 7261, 18136, 35761, 61576, 97021, ...

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

Columns k=0..3 give A000012, A143740, A125500, A362380.

Cf. A362378, A362394.

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

E.g.f. A_k(x) of column k satisfies A_k(x) = exp(x + k*x^2/2 * A_k(x)).
A_k(x) = exp(x - LambertW(-k*x^2/2 * exp(x))).
A_k(x) = -2 * LambertW(-k*x^2/2 * exp(x))/(k*x^2) for k > 0.

A362392 E.g.f. satisfies A(x) = exp(x + x^3 * A(x)).

Original entry on oeis.org

1, 1, 1, 7, 49, 241, 2041, 26041, 282913, 3449377, 57170161, 973059121, 16847893921, 343341027745, 7680743819113, 175958943331081, 4375517632543681, 118932887426911681, 3374685950589927649, 100735118425384221025, 3217474234925998764481

Offset: 0

Seiichi Manyama, Apr 20 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..414
Eric Weisstein's World of Mathematics, Lambert W-Function.

Column k=6 of A362378.

Cf. A118395, A125500, A362393, A362430.

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

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

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

A362490 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 * (3j+1)^(n-2j-1) / (j! * (n-3*j)!).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 17, 1, 1, 1, 1, 4, 33, 161, 1, 1, 1, 1, 5, 49, 321, 1351, 1, 1, 1, 1, 6, 65, 481, 2841, 12391, 1, 1, 1, 1, 7, 81, 641, 4471, 31641, 153385, 1, 1, 1, 1, 8, 97, 801, 6241, 57751, 498849, 2388905, 1

Offset: 0

Seiichi Manyama, Apr 22 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,   17,   33,   49,   65,   81,    97, ...
  1,  161,  321,  481,  641,  801,   961, ...
  1, 1351, 2841, 4471, 6241, 8151, 10201, ...

Winston de Greef, Table of n, a(n) for n = 0..11324 (150 antidiagonals)
Eric Weisstein's World of Mathematics, Lambert W-Function.

Columns k=0..3 give A000012, A362477, A362478, A362479.

Cf. A362378.

T(n, k) = n! * sum(j=0, n\3, (k/6)^j*(3*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)^3).
A_k(x) = exp(x - LambertW(-k*x^3/2 * exp(3*x))/3).
A_k(x) = ( -2 * LambertW(-k*x^3/2 * exp(3*x))/(k*x^3) )^(1/3) for k > 0.

A362381 E.g.f. satisfies A(x) = exp(x + x^3/6 * A(x)).

Original entry on oeis.org

1, 1, 1, 2, 9, 41, 191, 1191, 9353, 77897, 704861, 7352621, 85323921, 1058023825, 14155416003, 206100005931, 3217934262481, 53320102598481, 939087824434009, 17562552535939705, 346668611080774081, 7196193133818592961, 156944931623033340711

Offset: 0

Seiichi Manyama, Apr 20 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..447
Eric Weisstein's World of Mathematics, Lambert W-Function.

Column k=1 of A362378.

Cf. A190865, A362477.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x-lambertw(-x^3/6*exp(x)))))

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

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

A362390 E.g.f. satisfies A(x) = exp(x + x^3/3 * A(x)).

Original entry on oeis.org

1, 1, 1, 3, 17, 81, 441, 3641, 33825, 318753, 3505521, 45095601, 616484001, 9013086369, 145909533225, 2556431401161, 47388760825281, 937507626246081, 19840711661183457, 443937299529447009, 10456231167451597761, 259738234024404363201

Offset: 0

Seiichi Manyama, Apr 20 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..434
Eric Weisstein's World of Mathematics, Lambert W-Function.

Column k=2 of A362378.

Cf. A001470, A362478.

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

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

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

A362391 E.g.f. satisfies A(x) = exp(x + x^3/2 * A(x)).

Original entry on oeis.org

1, 1, 1, 4, 25, 121, 751, 7351, 73417, 749449, 9477181, 136883341, 2041250641, 33289802833, 608025141907, 11815916748091, 242532915013201, 5369303859003601, 126896359555326745, 3153096762426186553, 82705881733348530241, 2293511922269658189121

Offset: 0

Seiichi Manyama, Apr 20 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..427
Eric Weisstein's World of Mathematics, Lambert W-Function.

Column k=3 of A362378.

Cf. A362380, A362479.

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

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

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

cp's OEIS Frontend

A362377 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..floor(n/2)} (k/2)^j * (j+1)^(n-j-1) / (j! * (n-2*j)!).

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A362392 E.g.f. satisfies A(x) = exp(x + x^3 * A(x)).

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A362490 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 * (3*j+1)^(n-2*j-1) / (j! * (n-3*j)!).

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A362381 E.g.f. satisfies A(x) = exp(x + x^3/6 * A(x)).

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A362390 E.g.f. satisfies A(x) = exp(x + x^3/3 * A(x)).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A362391 E.g.f. satisfies A(x) = exp(x + x^3/2 * A(x)).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A362490 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 * (3j+1)^(n-2j-1) / (j! * (n-3*j)!).