A204858 -id:A204858 - cp's OEIS frontend

A204856 G.f.: Sum_{n>=0} x^(n(n+1)/2) / Product_{k=1..n} (1 - kx^k).

1, 1, 1, 2, 2, 4, 5, 9, 11, 22, 27, 49, 68, 115, 157, 279, 372, 628, 914, 1457, 2070, 3457, 4840, 7753, 11442, 17768, 25824, 41315, 59008, 92140, 137212, 208524, 305472, 477659, 691381, 1058019, 1575694, 2370618, 3491693, 5359888, 7796346, 11799263, 17583757

Offset: 0

Paul D. Hanna, Jan 20 2012

nonn

			G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 2*x^4 + 4*x^5 + 5*x^6 + 9*x^7 + 11*x^8 +...
where A(x) = 1 + x/(1-x) + x^3/((1-x)*(1-2*x^2)) + x^6/((1-x)*(1-2*x^2)*(1-3*x^3)) + x^10/((1-x)*(1-2*x^2)*(1-3*x^3)*(1-4*x^4)) +...

Seiichi Manyama, Table of n, a(n) for n = 0..6279 (terms 0..250 from Vaclav Kotesovec)

Cf. A204855, A204858.

Column sums of A367955.

Table[SeriesCoefficient[Sum[x^Binomial[n + 1, 2]/Product[(1 - k*x^k), {k, 1, n}], {x, 0, 100}], {x, 0, n}], {n, 0, 50}] (* G. C. Greubel, Dec 19 2017 *)

{a(n)=polcoeff(1+sum(m=1,n,x^(m*(m+1)/2)/prod(k=1,m,1-k*x^k+x*O(x^n))),n)}

G.f.: 1/(1 - x/(1 - x^2*(1-x)/(1-x^2 - x^3*(1-2*x^2)/(1-2*x^3 - x^4*(1-3*x^3)/(1-3*x^4 - x^5*(1-4*x^4)/(1-4*x^5 -...)))))), a continued fraction.
From Vaclav Kotesovec, Jun 18 2019: (Start)
a(n) ~ c * 3^(n/3), where
c = 23.5612420584121380174441491950859168338330954540437... if mod(n,3)=0
c = 23.5209031427848763179214171003561794127717213180726... if mod(n,3)=1
c = 23.5214569018665529984420312927586688667133017590049... if mod(n,3)=2
(End)

A204857 G.f.: Sum_{n>=0} n!x^(n(n+1)/2) / Product_{k=1..n} (1 - (n-k+1)*x^k).

Original entry on oeis.org

1, 1, 1, 3, 5, 11, 27, 61, 151, 375, 1001, 2699, 7635, 22069, 65695, 199671, 620417, 1966715, 6367323, 21059149, 71216311, 246322503, 871268465, 3148964147, 11613253707, 43625643373, 166606282471, 645633978279, 2534590357457, 10066575332603

Offset: 0

Paul D. Hanna, Jan 20 2012

nonn

			G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 5*x^4 + 11*x^5 + 27*x^6 + 61*x^7 +...
where A(x) = 1 + x/(1-x) + 2!*x^3/((1-2*x)*(1-x^2)) + 3!*x^6/((1-3*x)*(1-2*x^2)*(1-x^3)) + 4!*x^10/((1-4*x)*(1-3*x^2)*(1-2*x^3)*(1-x^4)) +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..250

Cf. A204858, A204855.

Table[SeriesCoefficient[Sum[n!*x^Binomial[n + 1, 2]/Product[(1 - (n - k + 1)*x^k), {k, 1, n}], {n, 0, 100}], {x, 0, n}], {n, 0, 50}] (* G. C. Greubel, Dec 19 2017 *)

{a(n)=polcoeff(1+sum(m=1,n,m!*x^(m*(m+1)/2)/prod(k=1,m,1-(m-k+1)*x^k+x*O(x^n))),n)}

A291969 Triangle read by rows: T(n,k) = k * (T(n-k,k-1) + T(n-k,k)) with T(0,0) = 1 for 0 <= k <= A003056(n).

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 1, 2, 0, 1, 2, 0, 1, 6, 0, 1, 6, 6, 0, 1, 14, 6, 0, 1, 14, 18, 0, 1, 30, 36, 0, 1, 30, 60, 24, 0, 1, 62, 96, 24, 0, 1, 62, 198, 72, 0, 1, 126, 270, 144, 0, 1, 126, 474, 336, 0, 1, 254, 780, 480, 120, 0, 1, 254, 1188, 1080, 120, 0, 1, 510, 1800

Offset: 0

Seiichi Manyama, Sep 07 2017

			First few rows are:
  1;
  0, 1;
  0, 1;
  0, 1,  2;
  0, 1,  2;
  0, 1,  6;
  0, 1,  6,  6;
  0, 1, 14,  6;
  0, 1, 14, 18;
  0, 1, 30, 36;
  0, 1, 30, 60, 24.

Seiichi Manyama, Rows n = 0..481, flattened

Row sums give A204858.
Columns 0-1 give A000007, A000012.
Cf. A291960.

G.f. of column k: k! * x^(k*(k+1)/2) / Product_{j=1..k} (1-j*x^j).

A306665 Expansion of Sum_{k>=0} k! * x^(k*(k+1)/2) / Product_{j=1..k} (1 - x^j)^j.

Original entry on oeis.org

1, 1, 1, 3, 3, 7, 13, 19, 31, 57, 99, 145, 253, 391, 661, 1071, 1647, 2617, 4189, 6439, 10183, 15999, 24195, 37537, 57553, 87925, 132841, 202899, 306147, 458827, 688501, 1030147, 1533535, 2280549, 3370947, 4986265, 7354573, 10779763, 15804901, 23102271, 33685239

Offset: 0

Ilya Gutkovskiy, Mar 04 2019

nonn

Cf. A032020, A204858, A206138.

nmax = 40; CoefficientList[Series[Sum[k! x^(k (k + 1)/2)/Product[(1 - x^j)^j, {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]

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A204856 G.f.: Sum_{n>=0} x^(n(n+1)/2) / Product_{k=1..n} (1 - kx^k).

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A204857 G.f.: Sum_{n>=0} n!x^(n(n+1)/2) / Product_{k=1..n} (1 - (n-k+1)*x^k).

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A291969 Triangle read by rows: T(n,k) = k * (T(n-k,k-1) + T(n-k,k)) with T(0,0) = 1 for 0 <= k <= A003056(n).

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A306665 Expansion of Sum_{k>=0} k! * x^(k*(k+1)/2) / Product_{j=1..k} (1 - x^j)^j.

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cp's OEIS Frontend

A204856 G.f.: Sum_{n>=0} x^(n*(n+1)/2) / Product_{k=1..n} (1 - k*x^k).

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A204857 G.f.: Sum_{n>=0} n!*x^(n*(n+1)/2) / Product_{k=1..n} (1 - (n-k+1)*x^k).

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PARI

A291969 Triangle read by rows: T(n,k) = k * (T(n-k,k-1) + T(n-k,k)) with T(0,0) = 1 for 0 <= k <= A003056(n).

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Author

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Formula

A306665 Expansion of Sum_{k>=0} k! * x^(k*(k+1)/2) / Product_{j=1..k} (1 - x^j)^j.

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Author

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Mathematica

A204856 G.f.: Sum_{n>=0} x^(n(n+1)/2) / Product_{k=1..n} (1 - kx^k).

A204857 G.f.: Sum_{n>=0} n!x^(n(n+1)/2) / Product_{k=1..n} (1 - (n-k+1)*x^k).