cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A301569 Expansion of Product_{k>=1} (1 + x^(5*k))*(1 + x^(5*k-2)).

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 0, 0, 2, 0, 1, 1, 0, 3, 0, 2, 2, 0, 5, 0, 2, 4, 0, 7, 1, 3, 7, 0, 10, 2, 4, 11, 0, 14, 4, 5, 17, 0, 19, 8, 6, 25, 1, 25, 13, 8, 36, 2, 33, 21, 10, 50, 4, 43, 33, 12, 69, 8, 55, 49, 15, 93, 14, 70, 71, 19, 124, 23, 88, 102, 24, 163, 37, 110, 142, 31
Offset: 0

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Author

Ilya Gutkovskiy, Mar 23 2018

Keywords

Comments

Number of partitions of n into distinct parts congruent to 0 or 3 mod 5.

Examples

			a(13) = 3 because we have [13], [10, 3] and [8, 5].

Links

Crossrefs

Cf. A035369, A036822, A047218, A203776, A219607, A281271, A301562, A301563, A301564, A301565, A301567, A301568, A301570.

Programs

  • Mathematica
    nmax = 75; CoefficientList[Series[Product[(1 + x^(5 k)) (1 + x^(5 k - 2)), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 75; CoefficientList[Series[x^2 QPochhammer[-1, x^5] QPochhammer[-x^(-2), x^5]/(2 (1 + x^2)), {x, 0, nmax}], x]
nmax = 75; CoefficientList[Series[Product[(1 + Boole[MemberQ[{0, 3}, Mod[k, 5]]] x^k), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=2} (1 + x^A047218(k)).
a(n) ~ exp(Pi*sqrt(2*n/15)) / (2^(37/20) * 15^(1/4) * n^(3/4)). - Vaclav Kotesovec, Mar 24 2018

A301570 Expansion of Product_{k>=1} (1 + x^(5*k))*(1 + x^(5*k-1)).

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 0, 0, 0, 2, 1, 0, 0, 1, 3, 2, 0, 0, 2, 5, 2, 0, 0, 4, 7, 3, 0, 1, 7, 10, 4, 0, 2, 11, 14, 5, 0, 4, 17, 19, 6, 0, 8, 25, 25, 8, 1, 13, 36, 33, 10, 2, 21, 50, 43, 12, 4, 33, 69, 55, 15, 8, 49, 93, 70, 18, 14, 71, 124, 88, 23, 23, 102, 163, 110, 29, 37
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 23 2018

Keywords

Comments

Number of partitions of n into distinct parts congruent to 0 or 4 mod 5.

Examples

			a(14) = 3 because we have [14], [10, 4] and [9, 5].

Links

Crossrefs

Cf. A035370, A036820, A047208, A203776, A219607, A281243, A301562, A301563, A301564, A301565, A301567, A301568, A301569.

Programs

  • Mathematica
    nmax = 76; CoefficientList[Series[Product[(1 + x^(5 k)) (1 + x^(5 k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 76; CoefficientList[Series[x QPochhammer[-1, x^5] QPochhammer[-x^(-1), x^5]/(2 (1 + x)), {x, 0, nmax}], x]
nmax = 76; CoefficientList[Series[Product[(1 + Boole[MemberQ[{0, 4}, Mod[k, 5]]] x^k), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=2} (1 + x^A047208(k)).
a(n) ~ exp(Pi*sqrt(2*n/15)) / (2^(41/20) * 15^(1/4) * n^(3/4)). - Vaclav Kotesovec, Mar 24 2018

A351634 G.f. A(x) satisfies: 1 + x = P(A(x)) / Q(A(x)), where P(x) = Product_{n>=0} (1 + x^(5*n+1))*(1 + x^(5*n+4)) and Q(x) = Product_{n>=0} (1 + x^(5*n+2))*(1 + x^(5*n+3)), with A(0) = 0, A'(0) = 1.

Original entry on oeis.org

1, 1, 4, 14, 59, 258, 1187, 5623, 27302, 135063, 678468, 3451272, 17742514, 92034588, 481112574, 2532013892, 13404579322, 71336740012, 381416582710, 2047875323729, 11036900422910, 59686672359369, 323788693045886, 1761507417706018
Offset: 1

Views

Author

Paul D. Hanna, Mar 14 2022

Keywords

Examples

			G.f.: A(x) = x + x^2 + 4*x^3 + 14*x^4 + 59*x^5 + 258*x^6 + 1187*x^7 + 5623*x^8 + 27302*x^9 + 135063*x^10 + 678468*x^11 + 3451272*x^12 + ...
where A = A(x) satisfies the infinite product:
1 + x = (1 + A)*(1 + A^4)/((1 + A^2)*(1 + A^3)) * (1 + A^6)*(1 + A^9)/((1 + A^7)*(1 + A^8)) * (1 + A^11)*(1 + A^14)/((1 + A^12)*(1 + A^13)) * (1 + A^16)*(1 + A^19)/((1 + A^17)*(1 + A^18)) * ...
equivalently,
1 + x = P(A(x)) / Q(A(x))
where
P(A(x)) = 1 + x + x^2 + 4*x^3 + 15*x^4 + 64*x^5 + 286*x^6 + 1332*x^7 + 6378*x^8 + 31224*x^9 + 155527*x^10 + ...
Q(A(x)) = 1 + x^2 + 3*x^3 + 12*x^4 + 52*x^5 + 234*x^6 + 1098*x^7 + 5280*x^8 + 25944*x^9 + 129583*x^10 + ...
and
P(x) = 1 + x + x^4 + x^5 + x^6 + x^7 + x^9 + 2*x^10 + 2*x^11 + x^12 + x^13 + 2*x^14 + ... + A203776(n)*x^n + ...
Q(x) = 1 + x^2 + x^3 + x^5 + x^7 + x^8 + x^9 + 2*x^10 + x^11 + 2*x^12 + 2*x^13 + x^14 + ... + A219607(n)*x^n + ...
also
P(x)/Q(x) = 1 + x - x^2 - 2*x^3 + x^4 + 3*x^5 + x^6 - 3*x^7 - 4*x^8 + x^9 + 6*x^10 + 3*x^11 - 6*x^12 - 8*x^13 + 2*x^14 + 12*x^15 + 6*x^16 - 11*x^17 - 15*x^18 + 3*x^19 + 22*x^20 + ...
such that A(x) = Series_Reversion(P(x)/Q(x) - 1).

Links

Crossrefs

Cf. A203776 (P), A219607 (Q).

Programs

  • Mathematica
    Rest[CoefficientList[InverseSeries[Series[-1 + QPochhammer[-x, x^5] * QPochhammer[-x^4, x^5] / (QPochhammer[-x^2, x^5] * QPochhammer[-x^3, x^5]), {x, 0, 25}], x], x]] (* Vaclav Kotesovec, Jan 17 2024 *)
(* Calculation of constant d: *) 1/r /. FindRoot[{1 + r == QPochhammer[-s, s^5]* QPochhammer[-s^4, s^5]/(QPochhammer[-s^2, s^5] * QPochhammer[-s^3, s^5]), 5*s^4*QPochhammer[-s^4, s^5]* Derivative[0, 1][QPochhammer][-s, s^5] + 1/s*QPochhammer[-s, s^5] * (-1/Log[s^5] * QPochhammer[-s^4, s^5]*(QPolyGamma[0, Log[-s]/Log[s^5], s^5] - 2*QPolyGamma[0, Log[-s^2]/Log[s^5], s^5] - 3*QPolyGamma[0, Log[-s^3]/Log[s^5], s^5] + 4*QPolyGamma[0, Log[-s^4]/Log[s^5], s^5]) - 5*s^5*QPochhammer[-s^4, s^5]*Derivative[0, 1][QPochhammer][ -s^2, s^5]/QPochhammer[-s^2, s^5] - 5*s^5*QPochhammer[-s^4, s^5] * Derivative[0, 1][QPochhammer][-s^3, s^5]/QPochhammer[-s^3, s^5] + 5*s^5*Derivative[0, 1][QPochhammer][-s^4, s^5]) == 0}, {r, 1/5}, {s, 1/3}, WorkingPrecision -> 80] (* Vaclav Kotesovec, Jan 17 2024 *)
  • PARI
    /* As Series Reversion of P(x)/Q(x) - 1 */
{a(n) = my(A=x,P,Q);
P = prod(m=0,n, (1 + x^(5*m+1))*(1 + x^(5*m+4)) +x*O(x^n));
Q = prod(m=0,n, (1 + x^(5*m+2))*(1 + x^(5*m+3)) +x*O(x^n));
polcoeff( serreverse( P/Q - 1 ), n)}
for(n=1,30,print1(a(n),", "))
  • PARI
    /* Obtain A(x) Using P(A(x))/Q(A(x)) = 1 + x */
{a(n) = my(A=[0,1],P,Q); for(i=1,n, A = concat(A,0);
PA = prod(m=0,#A, (1 + Ser(A)^(5*m+1))*(1 + Ser(A)^(5*m+4)) );
QA = prod(m=0,#A, (1 + Ser(A)^(5*m+2))*(1 + Ser(A)^(5*m+3)) );
A[#A] = -polcoeff( PA/QA ,#A-1) );A[n+1]}
for(n=1,30,print1(a(n),", "))

Formula

G.f.: A(x) = Series_Reversion( P(x)/Q(x) - 1 ), where P(x) = Product_{n>=0} (1 + x^(5*n+1))*(1 + x^(5*n+4)) and Q(x) = Product_{n>=0} (1 + x^(5*n+2))*(1 + x^(5*n+3)), with A(0) = 0, A'(0) = 1.
a(n) ~ c * d^n / n^(3/2), where d = 5.7997668905429653202956499676894864614337725024680731963895428378920947... and c = 0.0983896146762908218558422941662822756464709531976861748855671955... - Vaclav Kotesovec, Mar 15 2022
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