A361598 -id:A361598 - cp's OEIS frontend

A361600 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..n} binomial(n+(k-1)j,kj)/j!.

1, 1, 2, 1, 2, 5, 1, 2, 7, 16, 1, 2, 9, 34, 65, 1, 2, 11, 58, 209, 326, 1, 2, 13, 88, 473, 1546, 1957, 1, 2, 15, 124, 881, 4626, 13327, 13700, 1, 2, 17, 166, 1457, 10526, 52537, 130922, 109601, 1, 2, 19, 214, 2225, 20326, 145867, 677594, 1441729, 986410

Offset: 0

Seiichi Manyama, Mar 17 2023

			Square array begins:
    1,    1,    1,     1,     1,     1, ...
    2,    2,    2,     2,     2,     2, ...
    5,    7,    9,    11,    13,    15, ...
   16,   34,   58,    88,   124,   166, ...
   65,  209,  473,   881,  1457,  2225, ...
  326, 1546, 4626, 10526, 20326, 35226, ...

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

Columns k=0..3 give A000522, A002720, A361598, A361599.
Main diagonal gives A361607.
Cf. A293012.

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

E.g.f. of column k: exp( x/(1 - x)^k ) / (1-x).

T(n,k) = Sum_{j=0..n} (n+(k-1)*j)!/(k*j)! * binomial(n,j).

A343884 Expansion of e.g.f. exp( x/(1-x)^2 ) / (1-x)^2.

Original entry on oeis.org

1, 3, 15, 103, 885, 9051, 106843, 1425495, 21166953, 345678355, 6150501831, 118313311623, 2444917863325, 53982840948843, 1267645359117075, 31531781398100791, 827910838693667793, 22874802838645217955, 663243613324249850623, 20130710499843811837095

Offset: 0

Seiichi Manyama, Mar 18 2023

nonn

Winston de Greef, Table of n, a(n) for n = 0..430

Column k=2 of A361616.

Cf. A000262, A361598.

Table[n!*Sum[Binomial[n + k + 1, n - k]/k!, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 25 2023 *)

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x/(1-x)^2)/(1-x)^2))

a(n) = n! * sum(k=0, n, binomial(n+k+1,n-k)/k!) \\ Winston de Greef, Mar 19 2023

a(n) = n! * Sum_{k=0..n} binomial(n+k+1,n-k)/k! = Sum_{k=0..n} (n+k+1)!/(2*k+1)! * binomial(n,k).
From Vaclav Kotesovec, Mar 25 2023: (Start)
a(n) ~ exp(-1/12 + 3*2^(-2/3)*n^(2/3) - n) * n^(n + 1/2) / sqrt(6) * (1 + 2^(1/3)/n^(1/3) + 323/(360*2^(1/3)*n^(2/3))).
a(n) = 3*n*a(n-1) - 3*(n-1)^2*a(n-2) + (n-2)*(n-1)^2*a(n-3). (End)

A380664 Expansion of e.g.f. (1/x) * Series_Reversion( x * (1 - x) * exp(-x/(1 - x)^2) ).

Original entry on oeis.org

1, 2, 17, 268, 6277, 196416, 7716109, 365398496, 20271580137, 1290027358720, 92653747607401, 7414981595716608, 654373744057368493, 63136350047908917248, 6612064512998173129125, 747016321343021395603456, 90564758322246657646854481, 11727981253987656671672008704

Offset: 0

Seiichi Manyama, Jan 30 2025

nonn

Index entries for reversions of series

Cf. A377831, A380663.

Cf. A361598.

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

E.g.f. A(x) satisfies A(x) = exp(x * A(x)/(1 - x*A(x))^2)/(1 - x*A(x)).

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

A377595 E.g.f. satisfies A(x) = exp( x * A(x) / (1-x) ) / (1-x).

Original entry on oeis.org

1, 2, 11, 103, 1377, 24101, 523813, 13636463, 414246017, 14396807161, 563682761541, 24559156435595, 1178780540094193, 61810491468265541, 3515914378433242997, 215647516162031069191, 14187967957218808201089, 996767406049512569338481, 74478502236949781909301253

Offset: 0

Seiichi Manyama, Nov 14 2024

nonn

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

Cf. A362775, A377810.

Cf. A361598.

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-x/(1-x)^2))/(1-x)))

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

E.g.f.: exp( -LambertW(-x/(1-x)^2) )/(1-x).
a(n) = n! * Sum_{k=0..n} (k+1)^(k-1) * binomial(n+k,n-k)/k!.
a(n) ~ sqrt(1 + 2*exp(-1) - sqrt(1 + 4*exp(-1))) * sqrt(1 + 4*exp(-1) - sqrt(1 + 4*exp(-1))) * 2^(n + 3/2) * n^(n-1) / ((sqrt(1 + 4*exp(-1)) - 1)^(5/2) * exp(n) * (2 + exp(1) - exp(1/2)*sqrt(4 + exp(1)))^n). - Vaclav Kotesovec, Aug 05 2025

A373773 Expansion of e.g.f. exp(x^3 / (6 * (1 - x)^2)) / (1 - x).

Original entry on oeis.org

1, 1, 2, 7, 36, 240, 1930, 17990, 189840, 2233000, 28949200, 410009600, 6297999400, 104275571400, 1851050401200, 35065930299400, 705993054166400, 15051593241484800, 338705933426660800, 8021585392026606400, 199416162740963168000, 5191567315003621552000

Offset: 0

Seiichi Manyama, Jun 18 2024

nonn

Cf. A130906, A361597, A373772.
Cf. A361596, A361598.
Cf. A373757.

a(n) = n!*sum(k=0, n\3, binomial(n-k, n-3*k)/(6^k*k!));

a(n) = n! * Sum_{k=0..floor(n/3)} binomial(n-k,n-3*k)/(6^k * k!).
From Vaclav Kotesovec, Jun 18 2024: (Start)
Recurrence: 6*a(n) = 6*(3*n-2)*a(n-1) - 6*(n-1)*(3*n-4)*a(n-2) + 3*(n-2)*(n-1)*(2*n-3)*a(n-3) - (n-3)*(n-2)*(n-1)*a(n-4).
a(n) ~ 3^(-1/3) * exp(19/72 - 3^(-2/3)*n^(1/3) + 3^(2/3)*n^(2/3)/2 - n) * n^(n + 1/6). (End)

cp's OEIS Frontend

A361600 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..n} binomial(n+(k-1)*j,k*j)/j!.

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A343884 Expansion of e.g.f. exp( x/(1-x)^2 ) / (1-x)^2.

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A380664 Expansion of e.g.f. (1/x) * Series_Reversion( x * (1 - x) * exp(-x/(1 - x)^2) ).

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A377595 E.g.f. satisfies A(x) = exp( x * A(x) / (1-x) ) / (1-x).

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A373773 Expansion of e.g.f. exp(x^3 / (6 * (1 - x)^2)) / (1 - x).

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A361600 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..n} binomial(n+(k-1)j,kj)/j!.