A124504 -id:A124504 - cp's OEIS frontend

A343671 Number of partitions of an n-set without blocks of size 10.

1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115974, 678559, 4213465, 27643007, 190884307, 1382802389, 10478516523, 82847813908, 681895648039, 5830788687491, 51702731250650, 474630475600569, 4503991075480297, 44120379612630694, 445584481578266277, 4634070027874688433

Offset: 0

Ilya Gutkovskiy, Apr 25 2021

nonn

Cf. A000110, A000296, A027344, A097514, A124504, A343664, A343665, A343666, A343667, A343668, A343669.

a:= proc(n) option remember; `if`(n=0, 1, add(`if`(
       j=10, 0, a(n-j)*binomial(n-1, j-1)), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Jul 25 2023

nmax = 25; CoefficientList[Series[Exp[Exp[x] - 1 - x^10/10!], {x, 0, nmax}], x] Range[0, nmax]!
Table[n! Sum[(-1)^k BellB[n - 10 k]/((n - 10 k)! k! (10!)^k), {k, 0, Floor[n/10]}], {n, 0, 25}]
a[n_] := a[n] = If[n == 0, 1, Sum[If[k == 10, 0, Binomial[n - 1, k - 1]  a[n - k]], {k, 1, n}]]; Table[a[n], {n, 0, 25}]

E.g.f.: exp(exp(x) - 1 - x^10/10!).

a(n) = n! * Sum_{k=0..floor(n/10)} (-1)^k * Bell(n-10*k) / ((n-10*k)! * k! * (10!)^k).

A346272 Sum_{n>=0} a(n) * x^n / (n!)^2 = exp( BesselI(0,2*sqrt(x)) - 1 - x^3 / 36 ).

Original entry on oeis.org

1, 1, 3, 15, 115, 1196, 16282, 276158, 5713507, 140482000, 4047179258, 134447125418, 5097852537802, 218254682152053, 10469861372693621, 558373926672949031, 32908746221003292003, 2130712239317226923136, 150826951188229240683858, 11618459541824256750732794

Offset: 0

Ilya Gutkovskiy, Jul 12 2021

nonn

Cf. A023998, A061696, A124504, A346271.

nmax = 19; CoefficientList[Series[Exp[BesselI[0, 2 Sqrt[x]] - 1 - x^3/36], {x, 0, nmax}], x] Range[0, nmax]!^2
a[0] = a[1] = 1; a[n_] := a[n] = n a[n - 1] + n (n - 1)^2 a[n - 2]/2 + (1/n) Sum[Binomial[n, k]^2 k a[n - k], {k, 4, n}]; Table[a[n], {n, 0, 19}]

Sum_{n>=0} a(n) * x^n / (n!)^2 = exp( x + x^2 / 4 + Sum_{n>=4} x^n / (n!)^2 ).

a(0) = a(1) = 1; a(n) = n * a(n-1) + n * (n-1)^2 * a(n-2) / 2 + (1/n) * Sum_{k=4..n} binomial(n,k)^2 * k * a(n-k).

A361531 Expansion of e.g.f. exp(1 - exp(x) + x^3/6).

Original entry on oeis.org

1, -1, 0, 2, -3, -2, 21, -44, -62, 631, -1367, -3170, 34849, -86855, -302964, 3058342, -8509971, -36488802, 430842051, -1111575888, -6244999438, 78663444549, -250850311489, -1724880111306, 18475299723737, -65061274823853, -444914618968648, 6831921081061986

Offset: 0

Seiichi Manyama, Mar 14 2023

Cf. A293037, A337062.

Cf. A124504, A361489.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(1-exp(x)+x^3/6)))

a(0) = 1, a(1) = -1, a(2) = 0; a(n) = binomial(n-1,2) * a(n-3) - Sum_{k=1..n} binomial(n-1,k-1) * a(n-k).

cp's OEIS Frontend

A343671 Number of partitions of an n-set without blocks of size 10.

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A346272 Sum_{n>=0} a(n) * x^n / (n!)^2 = exp( BesselI(0,2*sqrt(x)) - 1 - x^3 / 36 ).

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A361531 Expansion of e.g.f. exp(1 - exp(x) + x^3/6).

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