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.

Showing 1-5 of 5 results.

A218153 G.f.: A(x) = exp( Sum_{n>=1} x^n/n * Product_{k>=1} (1 + x^(n*k)) ).

Original entry on oeis.org

1, 1, 2, 3, 6, 9, 16, 25, 42, 65, 105, 162, 256, 391, 605, 918, 1401, 2106, 3176, 4739, 7076, 10482, 15518, 22833, 33556, 49068, 71633, 104153, 151155, 218609, 315562, 454150, 652343, 934559, 1336328, 1906307, 2714409, 3856777, 5470236, 7743437, 10942743, 15435773
Offset: 0

Views

Author

Paul D. Hanna, Nov 01 2012

Keywords

Comments

Compare to the g.f. of A001383:
1 + x*exp( Sum_{n>=1} x^n/n * Product_{k>=1} 1/(1 - x^(n*k)) ).

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 6*x^4 + 9*x^5 + 16*x^6 + 25*x^7 +...
where
log(A(x)) = x/1*((1+x)*(1+x^2)*(1+x^3)*(1+x^4)*(1+x^5)*...) +
x^2/2*((1+x^2)*(1+x^4)*(1+x^6)*(1+x^8)*(1+x^10)*...) +
x^3/3*((1+x^3)*(1+x^6)*(1+x^9)*(1+x^12)*(1+x^15)*...) +
x^4/4*((1+x^4)*(1+x^8)*(1+x^12)*(1+x^16)*(1+x^20)*...) +
x^5/5*((1+x^5)*(1+x^10)*(1+x^15)*(1+x^20)*(1+x^25)*...) +...
Also, the g.f. is equal to the Euler transform of the distinct partitions A000009:
A(x) = 1/((1-x)^1*(1-x^2)^1*(1-x^3)^1*(1-x^4)^2*(1-x^5)^2*(1-x^6)^3*(1-x^7)^4*(1-x^8)^5*(1-x^9)^6*(1-x^10)^8*(1-x^11)^10*...*(1-x^n)^A000009(n-1)*...).

Links

Crossrefs

Cf. A218552, A218576, A000009, A001383, A089259.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Product[1/(1 - x^k)^PartitionsQ[k-1], {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 08 2016 *)
  • PARI
    {a(n)=polcoeff(exp(sum(m=1, n+1, x^m/m*prod(k=1, n\m+1, 1+x^(m*k)+x*O(x^n)))), n)}
for(n=0, 50, print1(a(n), ", "))

Formula

G.f.: Product_{n>=1} 1 / (1 - x^n)^A000009(n-1), where A000009(n) equals the number of distinct partitions of n. - Paul D. Hanna, Nov 16 2012

A218576 G.f.: exp( Sum_{n>=1} x^n/n * Product_{k>=1} (1 + x^(n*k)*(1 + x^k)^n) ).

Original entry on oeis.org

1, 1, 2, 4, 7, 14, 25, 44, 79, 137, 237, 408, 689, 1162, 1946, 3231, 5342, 8776, 14340, 23326, 37758, 60847, 97670, 156145, 248697, 394719, 624343, 984360, 1547187, 2424581, 3788730, 5904230, 9176723, 14226914, 22002523, 33947526, 52258177, 80268131, 123028407
Offset: 0

Views

Author

Paul D. Hanna, Nov 02 2012

Keywords

Comments

Compare to the dual g.f. of A219229:
exp( Sum_{n>=1} x^n/n * Product_{k>=1} (1 + x^(n*k)*(1 + x^n)^k) ).

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 7*x^4 + 14*x^5 + 25*x^6 + 44*x^7 +...
where
log(A(x)) = x/1*((1+x*(1+x))*(1+x^2*(1+x^2))*(1+x^3*(1+x^3))*...) +
x^2/2*((1+x^2*(1+x)^2)*(1+x^4*(1+x^2)^2)*(1+x^6*(1+x^3)^2)*...) +
x^3/3*((1+x^3*(1+x)^3)*(1+x^6*(1+x^2)^3)*(1+x^9*(1+x^3)^3)*...) +
x^4/4*((1+x^4*(1+x)^4)*(1+x^8*(1+x^2)^4)*(1+x^12*(1+x^3)^4)*...) +...
Explicitly,
log(A(x)) = x + 3*x^2/2 + 7*x^3/3 + 11*x^4/4 + 26*x^5/5 + 39*x^6/6 + 57*x^7/7 + 99*x^8/8 + 142*x^9/9 + 208*x^10/10 +...

Links

Crossrefs

Cf. A219229, A218575, A218552, A218153.

Programs

  • PARI
    {a(n)=polcoeff(exp(sum(m=1,n+1,x^m/m*prod(k=1,n\m,(1+x^(m*k)*(1+x^k+x*O(x^n))^m )))),n)}
for(n=0,50,print1(a(n),", "))

Formula

Conjecture: a(n) ~ c * d^n, where d = A060006 = 1.3247179572447... is the real root of the equation d*(d^2-1) = 1 and c = 43328430766.390... . - Vaclav Kotesovec, Apr 09 2016

A218551 G.f. satisfies: A(x) = exp( Sum_{n>=1} x^n/n * Product_{k>=1} 1/(1 - x^(n*k)*A(x^k)^n) ).

Original entry on oeis.org

1, 1, 2, 5, 13, 37, 106, 322, 987, 3119, 9985, 32499, 106910, 355524, 1191960, 4026739, 13689783, 46807685, 160842381, 555175377, 1923970425, 6691769948, 23351250882, 81729943060, 286842588316, 1009256119760, 3559337691300, 12579738946685, 44549347255523, 158058591860684
Offset: 0

Views

Author

Paul D. Hanna, Nov 01 2012

Keywords

Comments

Compare to the dual g.f. G(x) of A219231:
G(x) = exp( Sum_{n>=1} x^n/n * Product_{k>=1} 1/(1 - x^(n*k)*G(x^n)^k) ).

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 13*x^4 + 37*x^5 + 106*x^6 + 322*x^7 +...
where
log(A(x)) = x/(1*(1-x*A(x))*(1-x^2*A(x^2))*(1-x^3*A(x^3))*...) +
x^2/(2*(1-x^2*A(x)^2)*(1-x^4*A(x^2)^2)*(1-x^6*A(x^3)^2)*...) +
x^3/(3*(1-x^3*A(x)^3)*(1-x^6*A(x^2)^3)*(1-x^9*A(x^3)^3)*...) +
x^4/(4*(1-x^4*A(x)^4)*(1-x^8*A(x^2)^4)*(1-x^12*A(x^3)^4)*...) +...
Explicitly,
log(A(x)) = x + 3*x^2/2 + 10*x^3/3 + 31*x^4/4 + 106*x^5/5 + 342*x^6/6 + 1198*x^7/7 + 4071*x^8/8 + 14356*x^9/9 + 50408*x^10/10 +...

Crossrefs

Cf. A219231, A001383, A218552, A218575.

Programs

  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=exp(sum(m=1,n,x^m/m*prod(k=1,n\m+1,1/(1-x^(m*k)*subst(A,x,x^k +x*O(x^n))^m)))));polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

A219232 G.f. satisfies: A(x) = exp( Sum_{n>=1} x^n/n * Product_{k>=1} (1 + x^(n*k)*A(x^n)^k) ).

Original entry on oeis.org

1, 1, 2, 4, 11, 30, 92, 284, 918, 3005, 10043, 33943, 116138, 400862, 1395228, 4889389, 17240482, 61117789, 217709832, 778841527, 2797066886, 10080379573, 36444817306, 132147553180, 480444008087, 1751033068088, 6396352141777, 23414462628460, 85878613308907, 315556155264918
Offset: 0

Views

Author

Paul D. Hanna, Nov 16 2012

Keywords

Comments

Compare to the dual g.f. G(x) of A218552:
G(x) = exp( Sum_{n>=1} x^n/n * Product_{k>=1} (1 + x^(n*k)*G(x^k)^n) ).

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 11*x^4 + 30*x^5 + 92*x^6 + 284*x^7 +...
where
log(A(x)) = x/1*((1+x*A(x))*(1+x^2*A(x)^2)*(1+x^3*A(x)^3)*...) +
x^2/2*((1+x^2*A(x^2))*(1+x^4*A(x^2)^2)*(1+x^6*A(x^2)^3)*...) +
x^3/3*((1+x^3*A(x^3))*(1+x^6*A(x^3)^2)*(1+x^9*A(x^3)^3)*...) +
x^4/4*((1+x^4*A(x^4))*(1+x^8*A(x^4)^2)*(1+x^12*A(x^4)^3)*...) +...
Explicitly,
log(A(x)) = x + 3*x^2/2 + 7*x^3/3 + 27*x^4/4 + 86*x^5/5 + 321*x^6/6 + 1128*x^7/7 + 4163*x^8/8 + 15172*x^9/9 + 56078*x^10/10 +...

Crossrefs

Cf. A218552, A219229, A219231, A219230, A218153, A218153.

Programs

  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, x^m/m*prod(k=1, n\m+1, 1+x^(m*k)*subst(A, x, x^m +x*O(x^n))^k)))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

A219260 G.f. satisfies: A(x) = exp(Sum_{n>=1} x^n*A(x)^n/n * Product_{k>=1} (1 + x^(n*k)*A(x^k)^n)).

Original entry on oeis.org

1, 1, 3, 10, 37, 143, 576, 2393, 10178, 44133, 194341, 866867, 3908454, 17784385, 81562890, 376637216, 1749704080, 8171760933, 38346326963, 180707395127, 854850922373, 4057990958069, 19324260613400, 92288612451684, 441919933724974, 2121281845071105, 10205443975074195
Offset: 0

Views

Author

Paul D. Hanna, Nov 16 2012

Keywords

Comments

Compare to the dual g.f. G(x) of A219261:
G(x) = exp(Sum_{n>=1} x^n*G(x^n)/n * Product_{k>=1} (1+x^(n*k)*G(x^n)^k)).

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 10*x^3 + 37*x^4 + 143*x^5 + 576*x^6 + 2393*x^7 +...
where
log(A(x)) = x*A(x)/1*((1+x*A(x))*(1+x^2*A(x^2))*(1+x^3*A(x^3))*...) +
x^2*A(x)^2/2*((1+x^2*A(x)^2)*(1+x^4*A(x^2)^2)*(1+x^6*A(x^3)^2)*...) +
x^3*A(x)^3/3*((1+x^3*A(x)^3)*(1+x^6*A(x^2)^3)*(1+x^9*A(x^3)^3)*...) +
x^4*A(x)^4/4*((1+x^4*A(x)^4)*(1+x^8*A(x^2)^4)*(1+x^12*A(x^3)^4)*...) +...
Explicitly,
log(A(x)) = x + 5*x^2/2 + 22*x^3/3 + 101*x^4/4 + 461*x^5/5 + 2144*x^6/6 + 10109*x^7/7 + 48117*x^8/8 + 230998*x^9/9 + 1115875*x^10/10 +...

Crossrefs

Cf. A218552, A219262, A219261, A218153.

Programs

  • PARI
    {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, x^m*A^m/m*prod(k=1, n\m+1, 1+x^(m*k)*subst(A^m, x, x^k +x*O(x^n)))))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
Showing 1-5 of 5 results.

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