A275707 -id:A275707 - cp's OEIS frontend

A355501 Expansion of e.g.f. exp(3 * x * exp(x)).

1, 3, 15, 90, 633, 5028, 44217, 424434, 4399953, 48858984, 577372809, 7221983838, 95192539641, 1317190650636, 19071213218745, 288112248054882, 4530217559806497, 73976635012027344, 1252091246140278153, 21926952634345281030, 396671314081806278601

Offset: 0

Seiichi Manyama, Jul 04 2022

nonn

Column 3 of A187105.
Cf. A000248, A275707, A295623.
Cf. A351763.

With[{nn=20},CoefficientList[Series[Exp[3x Exp[x]],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jul 23 2025 *)

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

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

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=3*sum(j=1, i, j*binomial(i-1, j-1)*v[i-j+1])); v;

a(n) = sum(k=0, n, 3^k*k^(n-k)*binomial(n, k));

G.f.: Sum_{k>=0} (3 * x)^k/(1 - k*x)^(k+1).
a(0) = 1; a(n) = 3 * Sum_{k=1..n} k * binomial(n-1,k-1) * a(n-k).
a(n) = Sum_{k=0..n} 3^k * k^(n-k) * binomial(n,k).
a(n) ~ n^(n + 1/2) * exp(3*r*exp(r) - r/2 - n) / (sqrt(3*(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/3)/2). - Vaclav Kotesovec, Jul 06 2022

A187105 Triangle T(n,k) read by rows: number of height-2-restricted finite functions.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 10, 8, 3, 1, 41, 38, 15, 4, 1, 196, 216, 90, 24, 5, 1, 1057, 1402, 633, 172, 35, 6, 1, 6322, 10156, 5028, 1424, 290, 48, 7, 1, 41393, 80838, 44217, 13204, 2745, 450, 63, 8, 1, 293608, 698704, 424434, 134680, 28900, 4776, 658, 80, 9, 1

Offset: 1

Dennis P. Walsh, Mar 04 2011

Triangle T(n,k) with 1 <= k <= n+1 is the number of functions f:[n+1-k]->[n+1] such that f(f(f(x))) is undefined, that is, either f(x) or f(f(x)) is in {n+2-k,...,n+1}. Such functions are called height-2 restricted functions. Note that the null function, which occurs when k=n+1, vacuously satisfies the conditions for a height-2 restricted function, and hence T(n,n+1)=1. The sequence a(n)=T(n,1) is sequence A000248, the number of forests with n nodes and height at most 1. The height of a function f:D->C, with D a proper subset of finite C, is the maximum h such that (f^h)(x) exists for some x in D. A height restricted function f is acyclic since, if x is in a cycle of f, then (f^z)(x) exists for all positive integers z. [Note that [m] denotes the set of the first m positive integers and that f^m denotes the m-fold self-composition of f so that (f^0)(x)=x, (f^1)(x)=f(x),(f^2)(x)=f(f(x)), etc.]

			Triangle of initial terms:
     1
     1     1
     3     2     1
    10     8     3     1
    41    38    15     4     1
   196   216    90    24     5     1
  1057  1402   633   172    35     6     1
T(4,3) = 15 since there are 15 functions f:[2]->[5] such that either f(x) or f(f(x)) is in {3,4,5}. Using <f(1),f(2)> to denote these functions we have the following 15 functions: <2,3>, <2,4>, <2,5>, <3,1>, <3,3>, <3,4>, <3,5>, <4,1>, <4,3>, <4,4>, <4,5>, <5,1>, <5,3>, <5,4>, <5,5>.

Dennis Walsh, Notes on height-restricted finite functions

Cf. A000248, A275707, A355501.

seq(seq(sum(binomial(n+1-k,j)*k^j*j^(n+1-k-j),j=0..(n+1-k)),k=1..n),n=1..15); # triangle's right edge of ones is omitted with this program

t[n_, k_] := If[ k == n + 1, 1, Sum[ Binomial[n + 1 - k, j]*k^j*j^(n + 1 - k - j), {j, 0, n + 1 - k}]]; Table[ t[n, k], {n, 0, 9}, {k, n + 1}] // Flatten

T(n,k) = Sum_{j=0..n+1-k}binomial(n+1-k,j)*k^j*j^(n+1-k-j) for n>=0 and T(0,k) for k>=1.
E.g.f. of column k: exp(k*x*exp(x)).
With t(n,k) = T(n+k-1,k), t(n,k+j) = Sum_{i=0..n}binomial(n,i)*t(i,k)*t(n-i,j).

A295623 a(n) = n! * [x^n] exp(nxexp(x)).

Original entry on oeis.org

1, 1, 8, 90, 1424, 28900, 716292, 20972098, 708317248, 27108056808, 1159375192100, 54799938951934, 2836735081572240, 159606310760007436, 9698172715195196260, 632924646574215596850, 44153807025286701187328, 3278903858941755472870864, 258247909552273997037934788

Offset: 0

Ilya Gutkovskiy, Nov 24 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..360
N. J. A. Sloane, Transforms

Cf. A000248, A242817, A275707, A351765.

Table[n! SeriesCoefficient[Exp[n x Exp[x]], {x, 0, n}], {n, 0, 18}]
Table[Sum[BellY[n, k, n Range[n]], {k, 0, n}], {n, 0, 18}]

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

a(n) = n! * [x^n] exp(n*Sum_{k>=1} x^k/(k - 1)!).
From Seiichi Manyama, Jul 05 2022: (Start)
a(n) = [x^n] Sum_{k>=0} (n * x)^k/(1 - k*x)^(k+1).
a(n) = Sum_{k=0..n} n^k * k^(n-k) * binomial(n,k). (End)

A307996 Expansion of e.g.f. exp(1 - exp(x)(1 - 2x)).

Original entry on oeis.org

1, 1, 4, 15, 73, 410, 2591, 18165, 139266, 1155509, 10293729, 97815520, 986113613, 10499247005, 117603042220, 1381191356979, 16958788930317, 217132031279842, 2892337840164051, 40002168264724193, 573363461815952802, 8502905138072937073, 130268705062115090965, 2058969680487762098496

Offset: 0

Ilya Gutkovskiy, May 09 2019

nonn

Cf. A000248, A000354, A003727, A005408, A275707.

nmax = 23; CoefficientList[Series[Exp[1 - Exp[x] (1 - 2 x)], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[(2 k - 1) Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 23}]

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

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A355501 Expansion of e.g.f. exp(3 * x * exp(x)).

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A187105 Triangle T(n,k) read by rows: number of height-2-restricted finite functions.

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A295623 a(n) = n! * [x^n] exp(n*x*exp(x)).

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A307996 Expansion of e.g.f. exp(1 - exp(x)*(1 - 2*x)).

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A295623 a(n) = n! * [x^n] exp(nxexp(x)).

A307996 Expansion of e.g.f. exp(1 - exp(x)(1 - 2x)).