A356630 -id:A356630 - cp's OEIS frontend

A356628 a(n) = n! * Sum_{k=0..floor(n/2)} (n - 2k)^k/(n - 2k)!.

1, 1, 1, 7, 25, 181, 1561, 12811, 188497, 2071945, 38889361, 620762671, 12917838121, 291278938237, 6667342764265, 194869722610291, 5137978752994081, 177509783765281681, 5610285632192738977, 215195998789004395735, 8228064506323330305721

Offset: 0

Seiichi Manyama, Aug 18 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..366

Cf. A354436, A356629, A356630.

Cf. A216688, A356632, A358064.

a[n_] := n! * Sum[(n - 2*k)^k/(n - 2*k)!, {k, 0, Floor[n/2]}]; a[0] = 1; Array[a, 21, 0] (* Amiram Eldar, Aug 19 2022 *)

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

my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, x^k/(k!*(1-k*x^2)))))

E.g.f.: Sum_{k>=0} x^k / (k! * (1 - k*x^2)).

a(n) ~ sqrt(Pi) * exp((n-1)/(2*LambertW(exp(1/3)*(n-1)/3)) - 3*n/2) * n^((3*n + 1)/2) / (sqrt(1 + LambertW(exp(1/3)*(n - 1)/3)) * 3^((n+1)/2) * LambertW(exp(1/3)*(n-1)/3)^(n/2)). - Vaclav Kotesovec, Nov 01 2022

A356629 a(n) = n! * Sum_{k=0..floor(n/3)} (n - 3k)^k/(n - 3k)!.

Original entry on oeis.org

1, 1, 1, 1, 25, 121, 361, 5881, 82321, 547345, 6053041, 167991121, 2179469161, 22892967241, 788375451865, 18046198202761, 245523704069281, 7548055281543841, 270833271588545761, 5369819950838359585, 141456920470310708281, 6760255576117937586841

Offset: 0

Seiichi Manyama, Aug 18 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..393

Cf. A354436, A356628, A356630.

Cf. A354553, A356633, A358065.

a[n_] := n! * Sum[(n - 3*k)^k/(n - 3*k)!, {k, 0, Floor[n/3]}]; a[0] = 1; Array[a, 22, 0] (* Amiram Eldar, Aug 19 2022 *)

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

my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, x^k/(k!*(1-k*x^3)))))

E.g.f.: Sum_{k>=0} x^k / (k! * (1 - k*x^3)).

a(n) ~ sqrt(Pi/3) * exp((2*n - 3)/(6*LambertW(exp(1/4)*(2*n - 3)/8)) - 4*n/3) * n^(4*n/3 + 1/2) / (sqrt(1 + LambertW(exp(1/4)*(2*n - 3)/8)) * 2^(2*n/3 + 1/2) * LambertW(exp(1/4)*(2*n - 3)/8)^(n/3)). - Vaclav Kotesovec, Nov 01 2022

A356608 a(n) = n! * Sum_{k=0..floor(n/4)} (n - 4k)^k/(24^k (n - 4*k)!).

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 31, 106, 281, 1261, 13861, 106261, 558361, 2709136, 32802771, 447762316, 4093711441, 28011714641, 293624974441, 5549250905281, 80454378591121, 815886496908946, 8379058314620071, 168672787637953446, 3514729162490432041, 51656083670790267901

Offset: 0

Seiichi Manyama, Aug 18 2022

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..455

Cf. A354436, A356029, A356328.

Cf. A354552, A356630, A356634.

a[n_] := n! * Sum[(n - 4*k)^k/(24^k*(n - 4*k)!), {k, 0, Floor[n/4]}]; a[0] = 1; Array[a, 26, 0] (* Amiram Eldar, Aug 19 2022 *)

a(n) = n!*sum(k=0, n\4, (n-4*k)^k/(24^k*(n-4*k)!));

my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, x^k/(k!*(1-k*x^4/24)))))

E.g.f.: Sum_{k>=0} x^k / (k! * (1 - k*x^4/24)).

cp's OEIS Frontend

A356628 a(n) = n! * Sum_{k=0..floor(n/2)} (n - 2k)^k/(n - 2k)!.

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A356629 a(n) = n! * Sum_{k=0..floor(n/3)} (n - 3k)^k/(n - 3k)!.

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A356608 a(n) = n! * Sum_{k=0..floor(n/4)} (n - 4k)^k/(24^k (n - 4*k)!).

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cp's OEIS Frontend

A356628 a(n) = n! * Sum_{k=0..floor(n/2)} (n - 2*k)^k/(n - 2*k)!.

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A356629 a(n) = n! * Sum_{k=0..floor(n/3)} (n - 3*k)^k/(n - 3*k)!.

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A356608 a(n) = n! * Sum_{k=0..floor(n/4)} (n - 4*k)^k/(24^k * (n - 4*k)!).

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Author

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Programs

Mathematica

PARI

PARI

Formula

A356628 a(n) = n! * Sum_{k=0..floor(n/2)} (n - 2k)^k/(n - 2k)!.

A356629 a(n) = n! * Sum_{k=0..floor(n/3)} (n - 3k)^k/(n - 3k)!.

A356608 a(n) = n! * Sum_{k=0..floor(n/4)} (n - 4k)^k/(24^k (n - 4*k)!).