A015128 -id:A015128 - cp's OEIS frontend

A265835 Numbers n such that A015128(n)/2 is prime.

2, 4, 16, 36, 400, 1296, 1521, 52441

Offset: 1

Vaclav Kotesovec, Dec 16 2015

Next term, if it exists, is greater than 4000000. - Vaclav Kotesovec, updated Apr 12 2017

The values of a(n) are the squares of these integers for 1 < n < 9: 2, 4, 6, 20, 36, 39, 229. Squares also appear in the sequence of numbers k such that A015128(k)/2 is semiprime. - Altug Alkan, Dec 16 2015

			4 is a term because A015128(4)/2 = 14/2 = 7 is prime.

Cf. A014968, A015128, A035359, A046063, A299961.

Select[Range[2000], PrimeQ[Sum[PartitionsP[#-k]*PartitionsQ[k], {k, 0, #}]/2] &]

a015128(n) = polcoeff(exp(sum(m=1, n\2+1, 2*x^(2*m-1)/(1-x^(2*m-1)+x*O(x^n))/(2*m-1))), n);
for(n=1, 1e3, if(ispseudoprime(a015128(n)/2), print1(n, ", "))) \\ Altug Alkan, Dec 16 2015

A266497 Binomial transform of A015128.

Original entry on oeis.org

1, 3, 9, 27, 79, 225, 627, 1717, 4633, 12341, 32501, 84737, 218959, 561263, 1428287, 3610671, 9072367, 22668285, 56345835, 139382713, 343242533, 841713531, 2055944117, 5003148987, 12132552115, 29323810757, 70651867863, 169719163521, 406541986857, 971192810019

Offset: 0

Vaclav Kotesovec, Dec 30 2015

nonn

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

Cf. A015128, A218481, A266232, A294499.

A015128[n_]:=Sum[PartitionsP[n-k]*PartitionsQ[k], {k, 0, n}];
Table[Sum[Binomial[n, k]*A015128[k], {k, 0, n}], {n, 0, 30}]

a(n) ~ 2^(n-2) * exp(Pi*sqrt(n/2) + Pi^2/16) / n.

a(n) = [x^n] (1 + x)^n/theta_4(x), where theta_4() is the Jacobi theta function. - Ilya Gutkovskiy, Apr 20 2018

A277643 Partial sums of number of overpartitions (A015128).

Original entry on oeis.org

1, 3, 7, 15, 29, 53, 93, 157, 257, 411, 643, 987, 1491, 2219, 3259, 4731, 6793, 9657, 13605, 19005, 26341, 36245, 49533, 67261, 90789, 121855, 162679, 216087, 285655, 375903, 492527, 642671, 835283, 1081539, 1395347, 1793987, 2298873, 2936465, 3739401, 4747849

Offset: 0

Vaclav Kotesovec, Oct 25 2016

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A015128, A000070, A036469, A026906.

Cf. A211971, A002865, A087897.

Accumulate[Table[Sum[PartitionsP[n-k]*PartitionsQ[k], {k, 0, n}], {n, 0, 50}]]
nmax = 50; CoefficientList[Series[1/(1-x) * Product[(1 + x^k)/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 25 2017 *)

a(n) = Sum_{k=0..n} A015128(k).
a(n) ~ exp(Pi*sqrt(n))/(4*Pi*sqrt(n)) * (1 + Pi/(4*sqrt(n))).
G.f.: 1/(1-x) * Product_{k>=1} (1 + x^k) / (1 - x^k). - Vaclav Kotesovec, Mar 25 2017
G.f.: 1/((1 - x)*theta_4(x)), where theta_4() is the Jacobi theta function. - Ilya Gutkovskiy, Apr 20 2018

A303662 Decimal expansion of Sum_{k>=1} 1/A015128(k).

Original entry on oeis.org

1, 0, 5, 9, 2, 5, 4, 7, 1, 3, 6, 4, 9, 8, 7, 4, 5, 2, 2, 2, 2, 2, 6, 5, 6, 2, 0, 5, 8, 8, 4, 2, 8, 4, 1, 1, 4, 9, 6, 7, 3, 2, 1, 2, 3, 4, 2, 7, 9, 8, 2, 7, 2, 3, 5, 3, 2, 3, 3, 6, 6, 1, 5, 8, 0, 1, 3, 9, 7, 5, 3, 6, 2, 2, 1, 5, 2, 4, 0, 4, 3, 2, 7, 3, 1, 8, 9, 5, 9, 5, 4, 0, 0, 9, 6, 8, 4, 7, 5, 9, 9, 3, 9, 9, 3, 4

Offset: 1

Vaclav Kotesovec, Apr 28 2018

			1.059254713649874522222656205884284114967321234279827235323366158...

Vaclav Kotesovec, Table of n, a(n) for n = 1..1850

Cf. A078506, A237515.

A219430 Number of overpartitions of n^2; a(n) = A015128(n^2).

Original entry on oeis.org

1, 2, 14, 154, 2062, 31066, 504886, 8652402, 154208270, 2832526306, 53287424374, 1022143389578, 19924535352374, 393685747760714, 7869272950148382, 158875743754158098, 3235672769357219854, 66405081412501161442, 1372115409786911859502, 28524372351269271839610

Offset: 0

Paul D. Hanna, Nov 19 2012

nonn

Limit a(n+1)/a(n) = exp(Pi) = 23.14069263...
a(n) ~ (cosh(Pi*n) - sinh(Pi*n)/(Pi*n)) / (4*n^2), a "remarkable approximation" due to "Ramanujan's false statement" (see formula 12 in "Jagged partitions" link).
By definition of A015128, an overpartition of n^2 is an ordered sequence of nonincreasing integers that sum to n^2, where the first occurrence of each integer may be overlined (see Hirschhorn and Sellers link).

			G.f.: A(x) = 1 + 2*x + 14*x^2 + 154*x^3 + 2062*x^4 + 31066*x^5 + 504886*x^6 +...
It appears that the logarithmic derivative of the g.f. A(x),
A'(x)/A(x) = 2 + 24*x + 386*x^2 + 6832*x^3 + 128442*x^4 + 2505720*x^5 + 50153770*x^6 + 1022997344*x^7 + 21170657906*x^8 +...+ A219431(n+1)*x^n +...
is congruent to 2/(1-x^2) mod 4.

Vaclav Kotesovec, Table of n, a(n) for n = 0..730 (terms 0..180 from Paul D. Hanna)
J.-F. Fortin, P. Jacob and P. Mathieu, Jagged partitions
M. D. Hirschhorn and J. A. Sellers, AN INFINITE FAMILY OF OVERPARTITION CONGRUENCES MODULO 12

Cf. A219431, A015128, A195584.

Table[Sum[PartitionsP[n^2-k]*PartitionsQ[k], {k, 0, n^2}], {n, 0, 20}] (* Vaclav Kotesovec, Nov 28 2015 *)

/* Formula: a(n) = [x^(n^2)] 1 / theta_4(x) */
{a(n)=polcoeff(1/(1+2*sum(k=1,n,(-x)^(k^2))+x*O(x^(n^2))),n^2)}
for(n=0,20,print1(a(n),", "))

/* Formula: a(n) = -2*Sum_{k=1..n} (-1)^k * A015128(n^2-k^2) */
{A015128(n)=polcoeff(1/(1+2*sum(k=1, sqrtint(n+1), (-x)^(k^2))+x*O(x^(n))), n)}
{a(n)=if(n==0,1,-2*sum(k=1, n, (-1)^k*A015128(n^2-k^2)))}
for(n=0, 25, print1(a(n), ", "))

a(n) = -2*Sum_{k=1..n} (-1)^k * A015128(n^2-k^2) for n>0 with a(0)=1.
a(n) = [x^(n^2)] 1 / ( Sum_{m=-inf..inf} (-x)^(m^2) ).
a(n) = [x^(n^2)] 1 / theta_4(x).
a(n) = [x^(n^2)] eta(x^2) / eta(x)^2.
a(n) = [x^(n^2)] Product_{m>=1} (1 + x^m) / (1 - x^m).
a(n) = [x^(n^2)] Product_{m>=1} 1 / ( (1 - x^(2*m)) * (1 - x^(2*m-1))^2 ).
a(n) = [x^(n^2)] exp( Sum_{m>=1} 2*x^(2*m-1)/(1 - x^(2*m-1))/(2*m-1) ).
a(n) = [x^(n^2)] exp( Sum_{m>=1} (sigma(2*m) - sigma(m)) * x^m/m ).

A294499 Inverse binomial transform of the number of overpartitions (A015128).

Original entry on oeis.org

1, 1, 1, 1, -1, 3, -5, 7, -7, 3, 5, -9, -17, 129, -417, 977, -1809, 2591, -2317, -1061, 10485, -27983, 49165, -51319, -26861, 311455, -1011473, 2393275, -4643591, 7521265, -9694135, 7738137, 4976985, -38789975, 106112817, -215068927, 354515933, -464539803

Offset: 0

Vaclav Kotesovec, Nov 01 2017

sign

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A266497, A281425, A293467.

Table[Sum[(-1)^(n-k)*Binomial[n, k]*Sum[PartitionsP[k-j]*PartitionsQ[j], {j, 0, k}], {k, 0, n}], {n, 0, 50}]

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * A015128(k).

a(n) = [x^n] (1 - x)^n/theta_4(x), where theta_4() is the Jacobi theta function. - Ilya Gutkovskiy, Nov 03 2017

A299961 Numbers k such that k divides the number of overpartitions of k (A015128).

Original entry on oeis.org

1, 2, 12, 13, 22, 29, 88, 284, 370, 781, 1116, 1472, 1518, 1592, 2431, 2475, 2625, 3286, 5264, 6264, 6444, 7512, 7875, 9900, 22515, 30248, 30946, 31500, 32995, 41580, 69920, 112320, 126000, 140580, 142668, 166084, 166968, 225354, 232000, 272538, 290064, 312000

Offset: 1

Ilya Gutkovskiy, Feb 28 2018

nonn

			284 is in the sequence because A015128(284) = 42480456349401075392 is divisible by 284.

Vaclav Kotesovec, Table of n, a(n) for n = 1..68

Cf. A015128, A051177, A056848, A265835, A294086.

More terms from Vaclav Kotesovec, Mar 02 2018

A265015 a(n) = A015128(n)^n.

Original entry on oeis.org

1, 2, 16, 512, 38416, 7962624, 4096000000, 4398046511104, 10000000000000000, 48717667557975775744, 451730952053751361306624, 7982572438812891719395180544, 268637376395543538746286686601216, 16132732437821617561429013924830773248

Offset: 0

Vaclav Kotesovec, Dec 05 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..95

Cf. A015128, A133018, A156616, A265094.

nmax = 15; A015128 = Rest[CoefficientList[Series[Product[(1+x^k)/(1-x^k), {k,1,nmax}], {x,0,nmax}], x]]; Flatten[{1, Table[A015128[[n]]^n, {n,1,nmax}]}]

a(n) ~ exp(Pi*n^(3/2) - sqrt(n)/Pi - 1/(2*Pi^2)) / (8^n * n^n) * (1 - 1/(3*Pi^3*sqrt(n))).

A307945 Exponential convolution of A015128 with themselves.

Original entry on oeis.org

1, 4, 16, 64, 252, 968, 3616, 13120, 46432, 160772, 545856, 1821056, 5979520, 19350552, 61795968, 194964672, 608261628, 1878140024, 5743681784, 17408223328, 52320105080, 156011658272, 461763417056, 1357182242560, 3962591708576, 11497241014652

Offset: 0

Vaclav Kotesovec, May 07 2019

nonn

Robert Israel, Table of n, a(n) for n = 0..2994

Cf. A015128, A266497, A294499, A307755, A307756.

S:= series(1/JacobiTheta4(0,q),q,101):
f:= n -> add(binomial(n,k)*coeff(S,q,k)*coeff(S,q,n-k),k=0..n):
map(f, [$0..100]); # Robert Israel, May 08 2019

A015128[n_]:=Sum[PartitionsP[n-k]*PartitionsQ[k], {k, 0, n}]; Table[Sum[Binomial[n, k]*A015128[k]*A015128[n-k], {k, 0, n}], {n, 0, 25}]

a(n) = Sum_{k=0..n} binomial(n,k) * A015128(k) * A015128(n-k).

a(n) ~ 2^(n-4) * exp(Pi*sqrt(2*n)) / n^2.

A356289 a(n) = Sum_{k=0..n} binomial(2n, n-k) v(k), where v(k) is the number of overpartitions of n (A015128).

Original entry on oeis.org

1, 4, 18, 82, 372, 1676, 7500, 33358, 147570, 649722, 2848524, 12441434, 54155774, 235008672, 1016971480, 4389589484, 18902538548, 81222609020, 348308661820, 1490884718484, 6370468593732, 27176620756392, 115760526170340, 492386739902574, 2091554077819948, 8873225318953248

Offset: 0

Vaclav Kotesovec, Aug 02 2022

nonn

Cf. A015128, A266497, A356280, A356281, A356282, A356283, A356290.

Table[Sum[Sum[PartitionsP[k-j]*PartitionsQ[j], {j, 0, k}] * Binomial[2*n, n-k], {k, 0, n}], {n, 0, 30}]

a(n) ~ 2^(2*n - 7/6) * exp(3 * Pi^(4/3) * n^(1/3) / 2^(8/3)) / (sqrt(3) * Pi^(2/3) * n^(2/3)).

cp's OEIS Frontend

A265835 Numbers n such that A015128(n)/2 is prime.

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A266497 Binomial transform of A015128.

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A277643 Partial sums of number of overpartitions (A015128).

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A303662 Decimal expansion of Sum_{k>=1} 1/A015128(k).

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A219430 Number of overpartitions of n^2; a(n) = A015128(n^2).

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A294499 Inverse binomial transform of the number of overpartitions (A015128).

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A299961 Numbers k such that k divides the number of overpartitions of k (A015128).

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A265015 a(n) = A015128(n)^n.

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A307945 Exponential convolution of A015128 with themselves.

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A356289 a(n) = Sum_{k=0..n} binomial(2n, n-k) v(k), where v(k) is the number of overpartitions of n (A015128).