A351762 -id:A351762 - cp's OEIS frontend

A351761 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..n} k^(n-j) * (n-j)^j/j!.

1, 1, 0, 1, 1, 0, 1, 2, 4, 0, 1, 3, 12, 21, 0, 1, 4, 24, 102, 148, 0, 1, 5, 40, 279, 1160, 1305, 0, 1, 6, 60, 588, 4332, 16490, 13806, 0, 1, 7, 84, 1065, 11536, 84075, 281292, 170401, 0, 1, 8, 112, 1746, 25220, 282900, 1958058, 5598110, 2403640, 0

Offset: 0

Seiichi Manyama, Feb 18 2022

			Square array begins:
  1,    1,     1,     1,      1,      1, ...
  0,    1,     2,     3,      4,      5, ...
  0,    4,    12,    24,     40,     60, ...
  0,   21,   102,   279,    588,   1065, ...
  0,  148,  1160,  4332,  11536,  25220, ...
  0, 1305, 16490, 84075, 282900, 746525, ...

Columns k=0..3 give A000007, A006153, A351762, A351763.
Main diagonal gives A351765.
Cf. A292860, A351776.

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

T(n, k) = if(n==0, 1, k*n*sum(j=0, n-1, binomial(n-1, j)*T(j, k)));

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

T(0,k) = 1 and T(n,k) = k * n * Sum_{j=0..n-1} binomial(n-1,j) * T(j,k) for n > 0.

A275707 Number of partial functions f:{1,2,...,n}->{1,2,...,n} such that every element in the domain of definition of f is mapped to a fixed point or to an element that is undefined by f.

Original entry on oeis.org

1, 2, 8, 38, 216, 1402, 10156, 80838, 698704, 6498674, 64579284, 681642238, 7605025720, 89318058858, 1100376445564, 14176837311158, 190498308591264, 2663482511782114, 38667106019619748, 581765160424218606, 9055862445043643656, 145619330650420134362

Offset: 0

Geoffrey Critzer, Aug 06 2016

nonn

			G.f. = 1 + 2*x + 8*x^2 + 38*x^3 + 216*x^4 + 1402*x^5 + 10156*x^6 + ...
a(2) = 8 because there are 9 = A000169(3) partial functions on a set with 2 elements and all of them have the stated property except 1->2,2->1.

Alois P. Heinz, Table of n, a(n) for n = 0..524

Column 2 of A187105.

Cf. A000027, A000169, A000248, A351762, A355501.

a:= n-> add(binomial(n, k)*add(binomial(n-k, f)*
        (f+k)^(n-k-f), f=0..n-k), k=0..n):
seq(a(n), n=0..30);  # Alois P. Heinz, Aug 07 2016

nn = 20; Range[0, nn]! CoefficientList[Series[ Exp[z Exp[z]]^2, {z, 0, nn}], z]
Table[Sum[BellY[n, k, 2 Range[n]], {k, 0, n}], {n, 0, 20}] (* Vladimir Reshetnikov, Nov 09 2016 *)

x='x+O('x^33); Vec(serlaplace(exp(2*x*exp(x)))) \\ Joerg Arndt, Nov 10 2016

my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (2*x)^k/(1-k*x)^(k+1))) \\ Seiichi Manyama, Jul 04 2022

a(n) = sum(k=0, n, 2^k*k^(n-k)*binomial(n, k)); \\ Seiichi Manyama, Jul 04 2022

E.g.f.: A(x)^2 = exp(2*B(x)) where A(x) is the e.g.f. for A000248 and B(x) is the e.g.f. for A000027.
E.g.f.: exp(2*x*exp(x)). - Joerg Arndt, Nov 10 2016
a(0) = 1; a(n) = Sum_{k=1..n} 2*k*binomial(n-1,k-1)*a(n-k). - Ilya Gutkovskiy, Nov 24 2017
From Seiichi Manyama, Jul 04 2022: (Start)
G.f.: Sum_{k>=0} (2 * x)^k/(1 - k*x)^(k+1).
a(n) = Sum_{k=0..n} 2^k * k^(n-k) * binomial(n,k). (End)
a(n) ~ n^(n + 1/2) * exp(2*r*exp(r) - r/2 - n) / (sqrt(2*(1 + 3*r + r^2)) * r^(n + 1/2)), where r = 2*w - 1/(2*w) + 5/(8*w^2) - 19/(24*w^3) + 209/(192*w^4) - 763/(480*w^5) + 4657/(1920*w^6) - 6855/(1792*w^7) + 199613/(32256*w^8) + ... and w = LambertW(sqrt(n)/2^(3/2)). - Vaclav Kotesovec, Jul 06 2022

A368267 Expansion of e.g.f. exp(2x) / (1 - 2x*exp(x)).

Original entry on oeis.org

1, 4, 24, 206, 2344, 33322, 568420, 11312366, 257293872, 6583516946, 187173328444, 5853594770806, 199705444781512, 7381068971010074, 293787494031046740, 12528831526596461438, 569923490454177217120, 27545552296682691393058

Offset: 0

Seiichi Manyama, Dec 19 2023

Cf. A351762, A368236, A368268, A368269.

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

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

a(n) ~ n! / (4 * LambertW(1/2)^(n+2) * (LambertW(1/2) + 1)). - Vaclav Kotesovec, Dec 29 2023

A352251 Expansion of e.g.f. 1 / (1 - x * sinh(x)) (even powers only).

Original entry on oeis.org

1, 2, 28, 966, 62280, 6452650, 980531916, 205438870014, 56760128400016, 19994672935658322, 8746764024725937300, 4651991306703670964518, 2956156902003429777549144, 2212026607642404922284728826, 1925137044528752884360406444380, 1928103808741894922401976601295950

Offset: 0

Ilya Gutkovskiy, Mar 09 2022

nonn

Cf. A006153, A006154, A009233, A094088, A205571, A210672, A351762, A352250, A352252, A352253.

nmax = 30; Take[CoefficientList[Series[1/(1 - x Sinh[x]), {x, 0, nmax}], x] Range[0, nmax]!, {1, -1, 2}]
a[0] = 1; a[n_] := a[n] = 2 Sum[Binomial[2 n, 2 k] k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 15}]

my(x='x+O('x^40), v=Vec(serlaplace(1 /(1-x*sinh(x))))); vector(#v\2, k, v[2*k-1]) \\ Michel Marcus, Mar 10 2022

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

A351777 Expansion of e.g.f. 1/(1 + 2xexp(x)).

Original entry on oeis.org

1, -2, 4, -6, -8, 150, -972, 3682, 6256, -289746, 3300460, -21071622, -27876312, 3156947014, -53217341660, 494232431250, 175171749088, -113735274256290, 2613309376750812, -32653995355358678, 36013529538641560, 10227377502146048118, -305630239215263764076

Offset: 0

Seiichi Manyama, Feb 19 2022

sign

Column k=2 of A351776.

Cf. A351762.

With[{nn=30},CoefficientList[Series[1/(1+2x Exp[x]),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Feb 06 2024 *)

my(N=40, x='x+O('x^N)); Vec(serlaplace(1/(1+2*x*exp(x))))

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

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

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

a(0) = 1 and a(n) = -2 * n * Sum_{k=0..n-1} binomial(n-1,k) * a(k) for n > 0.

A368268 Expansion of e.g.f. exp(-x) / (1 - 2xexp(x)).

Original entry on oeis.org

1, 1, 9, 71, 817, 11599, 197881, 3938087, 89569761, 2291869727, 65159228521, 2037767466679, 69521938950289, 2569515452879855, 102274007835523161, 4361566914028222919, 198403133940750790081, 9589223805173365594687, 490729273233730201604809

Offset: 0

Seiichi Manyama, Dec 19 2023

Cf. A351762, A368236, A368267, A368269.

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

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

A368269 Expansion of e.g.f. exp(-2x) / (1 - 2x*exp(x)).

Original entry on oeis.org

1, 0, 8, 46, 584, 8138, 139252, 2770206, 63009648, 1612255186, 45837395564, 1433503025414, 48906419204392, 1807570412699322, 71946432680652324, 3068220235065662062, 139570141248903198944, 6745706553985526731682, 345212056986241161670876

Offset: 0

Seiichi Manyama, Dec 19 2023

Cf. A351762, A368236, A368267, A368268.

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

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

cp's OEIS Frontend

A351761 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..n} k^(n-j) * (n-j)^j/j!.

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A275707 Number of partial functions f:{1,2,...,n}->{1,2,...,n} such that every element in the domain of definition of f is mapped to a fixed point or to an element that is undefined by f.

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Maple

Mathematica

PARI

PARI

PARI

Formula

A368267 Expansion of e.g.f. exp(2*x) / (1 - 2*x*exp(x)).

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PARI

Formula

A352251 Expansion of e.g.f. 1 / (1 - x * sinh(x)) (even powers only).

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Mathematica

PARI

Formula

A351777 Expansion of e.g.f. 1/(1 + 2*x*exp(x)).

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Mathematica

PARI

PARI

PARI

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A368268 Expansion of e.g.f. exp(-x) / (1 - 2*x*exp(x)).

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PARI

Formula

A368269 Expansion of e.g.f. exp(-2*x) / (1 - 2*x*exp(x)).

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Programs

PARI

Formula

A368267 Expansion of e.g.f. exp(2x) / (1 - 2x*exp(x)).

A351777 Expansion of e.g.f. 1/(1 + 2xexp(x)).

A368268 Expansion of e.g.f. exp(-x) / (1 - 2xexp(x)).

A368269 Expansion of e.g.f. exp(-2x) / (1 - 2x*exp(x)).