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 1-6 of 6 results.

A192405 G.f. satisfies: A(x) = 1 + Sum_{n>=1} x^(n+1) * A(x)^n/(1 - x*A(x)^(2*n)).

Original entry on oeis.org

1, 0, 1, 2, 4, 11, 33, 99, 310, 1016, 3413, 11682, 40751, 144476, 519013, 1886311, 6928012, 25684055, 96020957, 361742039, 1372442092, 5241062187, 20136335035, 77806111700, 302259125863, 1180207733657, 4630733662020, 18254415188073, 72283753111667
Offset: 0

Views

Author

Paul D. Hanna, Jun 30 2011

Keywords

Comments

Related q-series identity:
Sum_{n>=1} z^n*y*q^n/(1-y*q^(2*n)) = Sum_{n>=1} y^n*z*q^(2*n-1)/(1-z*q^(2*n-1)); here q=A(x), y=x, z=x.

Examples

			G.f.: A(x) = 1 + x^2 + 2*x^3 + 4*x^4 + 11*x^5 + 33*x^6 + 99*x^7 +...
which satisfies the following relations:
A(x) = 1 + x^2*A(x)/(1-x*A(x)^2) + x^3*A(x)^2/(1-x*A(x)^4) + x^4*A(x)^3/(1-x*A(x)^6) +...
A(x) = 1 + x^2*A(x)/(1-x*A(x)) + x^3*A(x)^3/(1-x*A(x)^3) + x^4*A(x)^5/(1-x*A(x)^5) +...
A(x) = 1 + x^2*A(x) + x^3*A(x)^3*(1 + 1/A(x)) + x^4*A(x)^6*(1 + 1/A(x) + 1/A(x)^3) + x^5*A(x)^10*(1 + 1/A(x) + 1/A(x)^3 + 1/A(x)^6) +...

Crossrefs

Cf. A192404, A192406, A192399, A192401.

Programs

  • PARI
    {a(n)=local(A=1+x^2);for(i=1,n,A=1+x*sum(m=1,n,x^m*A^m/(1-x*A^(2*m)+x*O(x^n))));polcoeff(A,n)}
  • PARI
    {a(n)=local(A=1+x^2);for(i=1,n,A=1+x*sum(m=1,n,x^m*A^(2*m-1)/(1-x*A^(2*m-1)+x*O(x^n))));polcoeff(A,n)}
  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=1+sum(m=1,n,x^(m+1)*A^(m*(m+1)/2)*sum(k=0,m-1,(A+x*O(x^n))^(-k*(k+1)/2) ) ) );polcoeff(A,n)}

Formula

G.f. satisfies: A(x) = 1 + Sum_{n>=1} x^(n+1)*A(x)^(2*n-1)/(1 - x*A(x)^(2*n-1)).
G.f. satisfies: A(x) = 1 + Sum_{n>=1} x^(n+1)*A(x)^(n*(n+1)/2) * Sum_{k=0..n-1} A(x)^(-k*(k+1)/2).
Equals the antidiagonal sums of square array A192404.

A192400 G.f. A(x) satisfies A(x) = 1 + Sum_{n>=1} A(x)^n * x^(2*n-1)/(1 - x^(2*n-1)).

Original entry on oeis.org

1, 1, 2, 5, 11, 26, 64, 158, 399, 1027, 2675, 7052, 18788, 50487, 136711, 372687, 1021942, 2816873, 7800510, 21691134, 60543553, 169561453, 476351239, 1342002198, 3790565335, 10732246631, 30453309502, 86589559266, 246672752090
Offset: 0

Views

Author

Paul D. Hanna, Jun 30 2011

Keywords

Comments

Related q-series identity:
Sum_{n>=1} y^n*z*q^(2*n-1)/(1-z*q^(2*n-1)) = Sum_{n>=1} z^n*y*q^n/(1-y*q^(2*n)); here q=x, y=A(x), z=1.

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 11*x^4 + 26*x^5 + 64*x^6 +...
which satisfies the following relations:
A(x) = 1 + A(x)*x/(1-x) + A(x)^2*x^3/(1-x^3) + A(x)^3*x^5/(1-x^5) +...
A(x) = 1 + A(x)*x/(1-A(x)*x^2) + A(x)*x^2/(1-A(x)*x^4) + A(x)*x^3/(1-A(x)*x^6) +...

Crossrefs

Cf. A192401, A192402.

Programs

  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=1+sum(m=1,n,A^m*x^(2*m-1)/(1-x^(2*m-1)+x*O(x^n))));polcoeff(A,n)}
  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=1+sum(m=1,n,A*x^m/(1-A*x^(2*m)+x*O(x^n))));polcoeff(A,n)}

Formula

G.f. satisfies: A(x) = 1 + Sum_{n>=1} A(x)*x^n/(1 - A(x)*x^(2*n)).

A192403 G.f. A(x) satisfies A(x) = 1 + Sum_{n>=1} A(x)^n * 2*x^n/(1 - 2*x^(2*n)).

Original entry on oeis.org

1, 2, 6, 26, 106, 474, 2210, 10638, 52578, 265286, 1360702, 7074030, 37191694, 197398394, 1056255758, 5691813546, 30860701490, 168236407482, 921576598970, 5070138584230, 28002574339634, 155204886300414, 862985636296302, 4812513873922710
Offset: 0

Views

Author

Paul D. Hanna, Jun 30 2011

Keywords

Comments

Related q-series identity:
Sum_{n>=1} z^n*y*q^n/(1-y*q^(2*n)) = Sum_{n>=1} y^n*z*q^(2*n-1)/(1-z*q^(2*n-1)); here q=x, y=2, z=A(x).

Examples

			G.f.: A(x) = 1 + 2*x + 6*x^2 + 26*x^3 + 106*x^4 + 474*x^5 + 2210*x^6 +...
which satisfies the following relations:
A(x) = 1 + A(x)*2*x/(1-2*x^2) + A(x)^2*2*x^2/(1-2*x^4) + A(x)^3*2*x^3/(1-2*x^6) +...
A(x) = 1 + 2*A(x)*x/(1-A(x)*x) + 4*A(x)*x^3/(1-A(x)*x^3) + 8*A(x)*x^5/(1-A(x)*x^5) +...

Crossrefs

Cf. A192401, A192402.

Programs

  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=1+sum(m=1,n,A^m*2*x^m/(1-2*x^(2*m)+x*O(x^n))));polcoeff(A,n)}
  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=1+sum(m=1,n,2^m*A*x^(2*m-1)/(1-A*x^(2*m-1)+x*O(x^n))));polcoeff(A,n)}

Formula

G.f. satisfies: A(x) = 1 + Sum_{n>=1} 2^n*A(x)*x^(2*n-1)/(1 - A(x)*x^(2*n-1)).

A307396 G.f. A(x) satisfies: A(x) = 1 + Sum_{k>=1} x^k*A(x)^k/(1 + x^k).

Original entry on oeis.org

1, 1, 1, 4, 9, 25, 78, 235, 734, 2355, 7637, 25096, 83394, 279563, 944559, 3213254, 10996236, 37829956, 130759164, 453879479, 1581472334, 5529435704, 19393856909, 68217376618, 240586328527, 850553637256, 3013750513593, 10700805837614, 38068482070675, 135674217800041
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 07 2019

Keywords

Examples

			G.f.: A(x) = 1 + x + x^2 + 4*x^3 + 9*x^4 + 25*x^5 + 78*x^6 + 235*x^7 + 734*x^8 + 2355*x^9 + 7637*x^10 + ...

Crossrefs

Cf. A048272, A192207, A192401, A307398, A307400.

Programs

  • Mathematica
    terms = 30; A[] = 0; Do[A[x] = 1 + Sum[x^k A[x]^k /(1 + x^k), {k, 1, j}] + O[x]^j, {j, 1, terms}]; CoefficientList[A[x], x]
terms = 30; A[] = 0; Do[A[x] = 1 + Sum[x^k Sum[(-1)^(k/d + 1) A[x]^d, {d, Divisors[k]}], {k, 1, j}] + O[x]^j, {j, 1, terms}]; CoefficientList[A[x], x]

Formula

G.f. A(x) satisfies: A(x) = 1 + Sum_{k>=1} x^k * Sum_{d|k} (-1)^(k/d+1)*A(x)^d.

A307400 G.f. A(x) satisfies: A(x) = 1 + Sum_{k>=1} k*x^k*A(x)^k/(1 + x^k).

Original entry on oeis.org

1, 1, 2, 9, 28, 109, 440, 1790, 7537, 32300, 140438, 618608, 2753510, 12366672, 55973926, 255059808, 1169143476, 5387268256, 24940059514, 115943355422, 541047868905, 2533458659581, 11900017205866, 56055896316345, 264748474342341, 1253414056154014, 5947373587731308
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 07 2019

Keywords

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 9*x^3 + 28*x^4 + 109*x^5 + 440*x^6 + 1790*x^7 + 7537*x^8 + 32300*x^9 + 140438*x^10 + ...

Crossrefs

Cf. A000593, A192207, A192401, A307396, A307398.

Programs

  • Mathematica
    terms = 27; A[] = 0; Do[A[x] = 1 + Sum[k x^k A[x]^k/(1 + x^k), {k, 1, j}] + O[x]^j, {j, 1, terms}]; CoefficientList[A[x], x]
terms = 27; A[] = 0; Do[A[x] = 1 + Sum[x^k Sum[(-1)^(k/d + 1) d A[x]^d, {d, Divisors[k]}], {k, 1, j}] + O[x]^j, {j, 1, terms}]; CoefficientList[A[x], x]

Formula

G.f. A(x) satisfies: A(x) = 1 + Sum_{k>=1} x^k * Sum_{d|k} (-1)^(k/d+1)*d*A(x)^d.

A192399 G.f. A(x) satisfies: A(x) = 1 + Sum_{n>=1} x^n * A(x)^n/(1 - x*A(x)^(2*n)).

Original entry on oeis.org

1, 1, 3, 11, 48, 233, 1218, 6722, 38668, 229864, 1403618, 8766186, 55818141, 361499355, 2376956264, 15845876429, 106988044753, 731026642533, 5051920683481, 35296182297157, 249249589433312, 1778775804736254, 12828718640894604
Offset: 0

Views

Author

Paul D. Hanna, Jun 30 2011

Keywords

Comments

Related q-series identity:
Sum_{n>=1} z^n*y*q^n/(1-y*q^(2*n)) = Sum_{n>=1} y^n*z*q^(2*n-1)/(1-z*q^(2*n-1)); here q=A(x), y=x, z=x.

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 11*x^3 + 48*x^4 + 233*x^5 + 1218*x^6 +...
which satisfies the following relations:
A(x) = 1 + x*A(x)/(1-x*A(x)^2) + x^2*A(x)^2/(1-x*A(x)^4) + x^3*A(x)^3/(1-x*A(x)^6) +...
A(x) = 1 + x*A(x)/(1-x*A(x)) + x^2*A(x)^3/(1-x*A(x)^3) + x^3*A(x)^5/(1-x*A(x)^5) +...
A(x) = 1 + x*A(x) + x^2*A(x)^3*(1 + 1/A(x)) + x^3*A(x)^6*(1 + 1/A(x) + 1/A(x)^3) + x^4*A(x)^10*(1 + 1/A(x) + 1/A(x)^3 + 1/A(x)^6) +...

Crossrefs

Cf. A192400, A192401, A192405.

Programs

  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=1+sum(m=1,n,x^m*A^m/(1-x*A^(2*m)+x*O(x^n))));polcoeff(A,n)}
  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=1+sum(m=1,n,x^m*A^(2*m-1)/(1-x*A^(2*m-1)+x*O(x^n))));polcoeff(A,n)}
  • PARI
    {a(n)=local(A=1+x);for(i=1,n,A=1+sum(m=1,n,x^m*A^(m*(m+1)/2)*sum(k=0,m-1,(A+x*O(x^n))^(-k*(k+1)/2) ) ) );polcoeff(A,n)}

Formula

G.f. satisfies: A(x) = 1 + Sum_{n>=1} x^n*A(x)^(2*n-1)/(1 - x*A(x)^(2*n-1)).
G.f. satisfies: A(x) = 1 + Sum_{n>=1} x^n*A(x)^(n*(n+1)/2) * Sum_{k=0..n-1} A(x)^(-k*(k+1)/2). - Paul D. Hanna, Jul 01 2011
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.