A294501 -id:A294501 - cp's OEIS frontend

A294500 Binomial transform of the number of planar partitions (A000219).

1, 2, 6, 19, 60, 185, 559, 1662, 4875, 14134, 40564, 115370, 325465, 911355, 2534595, 7004827, 19246626, 52596377, 143006632, 386984573, 1042537831, 2796803110, 7473161196, 19893461042, 52767059608, 139488323734, 367540167625, 965445514862, 2528516552660

Offset: 0

Vaclav Kotesovec, Nov 01 2017

nonn

Let 0 < p < 1, r > 0, v > 0, f(n) = v*exp(r*n^p)/n^b, then
Sum_{k=0..n} binomial(n,k) * f(k) ~ f(n/2) * 2^n * exp(g(n)), where
g(n) = p^2 * r^2 * n^p / (2^(1+2*p)*n^(1-p) + p*r*(1-p)*2^(1+p)).
Special cases:
p < 1/2, g(n) = 0
p = 1/2, g(n) = r^2/16
p = 2/3, g(n) = r^2 * n^(1/3) / (9 * 2^(1/3)) - r^3/81
p = 3/4, g(n) = 9*r^2*sqrt(n)/(64*sqrt(2)) - 27*r^3*n^(1/4)/(2048*2^(1/4)) + 81*r^4/65536
p = 3/5, g(n) = 9*r^2*n^(1/5)/(100*2^(1/5))
p = 4/5, g(n) = 2^(7/5)*r^2*n^(3/5)/25 - 4*2^(3/5)*r^3*n^(2/5)/625 + 8*2^(4/5)*r^4*n^(1/5)/15625 - 32*r^5/390625

Vaclav Kotesovec, Table of n, a(n) for n = 0..2930

Cf. A218481, A266232, A294501, A294502, A294504.

nmax = 40; s = CoefficientList[Series[Product[1/(1-x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x]; Table[Sum[Binomial[n, k] * s[[k+1]], {k, 0, n}], {n, 0, nmax}]

a(n) = Sum_{k=0..n} binomial(n,k) * A000219(k).
a(n) ~ exp(1/12 + 3 * Zeta(3)^(1/3) * n^(2/3) / 2^(4/3) + Zeta(3)^(2/3) * n^(1/3) / 2^(5/3) - Zeta(3)/12) * 2^(n + 7/18) * Zeta(3)^(7/36) / (A * sqrt(3*Pi) * n^(25/36)), where A is the Glaisher-Kinkelin constant A074962.
G.f.: (1/(1 - x))*exp(Sum_{k>=1} sigma_2(k)*x^k/(k*(1 - x)^k)). - Ilya Gutkovskiy, Aug 20 2018

A294503 Inverse binomial transform of A026007.

Original entry on oeis.org

1, 0, 1, 1, -3, 10, -23, 48, -92, 171, -321, 626, -1265, 2576, -5099, 9478, -15925, 22617, -21816, -8506, 121659, -436121, 1204710, -2962759, 6860591, -15427559, 34323613, -76269455, 169591278, -376162414, 827819644, -1798045927, 3839392935, -8041078328

Offset: 0

Vaclav Kotesovec, Nov 01 2017

sign

Vaclav Kotesovec, Table of n, a(n) for n = 0..3000

Cf. A026007, A294501, A294502, A294505.

nmax = 40; s = CoefficientList[Series[Product[(1+x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x]; Table[Sum[(-1)^(n-k) * Binomial[n, k] * s[[k+1]], {k, 0, n}], {n, 0, nmax}]

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * A026007(k).

G.f.: (1/(1 + x))*Product_{k>=1} (1 + x^k/(1 + x)^k)^k. - Ilya Gutkovskiy, Aug 20 2018

A294505 Inverse binomial transform of A156616.

Original entry on oeis.org

1, 1, 3, 3, 3, 7, -3, 13, -5, -7, 49, -97, 93, 155, -997, 2893, -5989, 9007, -7121, -10805, 63305, -169375, 321137, -418503, 152653, 1142657, -4565939, 11378145, -21893565, 32887315, -33140953, -1985517, 113177979, -348817177, 734074637, -1210600023

Offset: 0

Vaclav Kotesovec, Nov 01 2017

sign

Vaclav Kotesovec, Table of n, a(n) for n = 0..3000

Cf. A156616, A294501, A294503, A294504.

nmax = 40; s = CoefficientList[Series[Product[((1+x^k)/(1-x^k))^k, {k, 1, nmax}], {x, 0, nmax}], x]; Table[Sum[(-1)^(n-k) * Binomial[n, k] * s[[k+1]], {k, 0, n}], {n, 0, nmax}]

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * A156616(k).

G.f.: (1/(1 + x))*exp(Sum_{k>=1} (sigma_2(2*k) - sigma_2(k))*x^k/(2*k*(1 + x)^k)). - Ilya Gutkovskiy, Oct 15 2018

A307258 Expansion of (1/(1 + x)) * Product_{k>=1} 1/(1 - k*x^k/(1 + x)^k).

Original entry on oeis.org

1, 0, 2, -1, 5, -11, 36, -107, 311, -850, 2208, -5519, 13566, -33562, 84937, -220307, 579413, -1522616, 3954016, -10100863, 25416877, -63324271, 157248035, -391478354, 980410093, -2470810086, 6253495883, -15846525758, 40093721908, -101116823798, 254093749587, -636547773777

Offset: 0

Ilya Gutkovskiy, Apr 01 2019

sign

Inverse binomial transform of A006906.

Cf. A006906, A281425, A294501, A307260, A318127.

a:=series((1/(1+x))*mul(1/(1-k*x^k/(1+x)^k),k=1..100),x=0,32): seq(coeff(a,x,n),n=0..31); # Paolo P. Lava, Apr 03 2019

nmax = 31; CoefficientList[Series[1/(1 + x) Product[1/(1 - k x^k/(1 + x)^k), {k, 1, nmax}], {x, 0, nmax}], x]
Table[Sum[(-1)^(n - k) Binomial[n, k] Total[Times @@@ IntegerPartitions[k]], {k, 0, n}], {n, 0, 31}]

a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n,k)*A006906(k).

cp's OEIS Frontend

A294500 Binomial transform of the number of planar partitions (A000219).

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A294503 Inverse binomial transform of A026007.

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A294505 Inverse binomial transform of A156616.

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A307258 Expansion of (1/(1 + x)) * Product_{k>=1} 1/(1 - k*x^k/(1 + x)^k).

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