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.

A346627 G.f. A(x) satisfies: A(x) = 1 / (1 + x) + x * A(x)^3.

Original entry on oeis.org

1, 0, 1, 2, 7, 23, 82, 300, 1129, 4334, 16914, 66899, 267586, 1080516, 4398850, 18035084, 74402361, 308624282, 1286428765, 5385578256, 22635057148, 95471113565, 403983783772, 1714494024947, 7295949019114, 31124885587680, 133085594104222, 570266646942488
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 25 2021

Keywords

Comments

Inverse binomial transform of A200753.

Links

Crossrefs

Cf. A001764, A005043, A199475, A200753, A346628, A349299, A349300, A349301, A349302, A349303.

Programs

  • Mathematica
    nmax = 27; A[] = 0; Do[A[x] = 1/(1 + x) + x A[x]^3 + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[n_] := a[n] = (-1)^n + Sum[Sum[a[i] a[j] a[n - i - j - 1], {j, 0, n - i - 1}], {i, 0, n - 1}]; Table[a[n], {n, 0, 27}]
Table[Sum[(-1)^(n - k) Binomial[n + k, n - k] Binomial[3 k, k]/(2 k + 1), {k, 0, n}], {n, 0, 27}]

Formula

G.f.: Sum_{k>=0} ( binomial(3*k,k) / (2*k + 1) ) * x^k / (1 + x)^(2*k+1).
a(n) = (-1)^n + Sum_{i=0..n-1} Sum_{j=0..n-i-1} a(i) * a(j) * a(n-i-j-1).
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n+k,n-k) * binomial(3*k,k) / (2*k + 1).
a(n) ~ sqrt(198 + 38*sqrt(33)) * (19 + 3*sqrt(33))^n / (9 * sqrt(Pi) * n^(3/2) * 2^(3*n + 3)). - Vaclav Kotesovec, Jul 30 2021

A349299 G.f. A(x) satisfies: A(x) = 1 / ((1 + x) * (1 - x * A(x)^3)).

Original entry on oeis.org

1, 0, 1, 3, 13, 57, 265, 1273, 6281, 31634, 161985, 840795, 4413937, 23395376, 125028541, 672958929, 3644868105, 19850452482, 108639736105, 597190462341, 3295737021241, 18253432801480, 101425665527825, 565249069709027, 3158734991846065
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 13 2021

Keywords

Links

Crossrefs

Cf. A002293, A005043, A346627, A346664, A349289, A349300, A349301, A349302, A349303.

Programs

  • Mathematica
    nmax = 24; A[] = 0; Do[A[x] = 1/((1 + x) (1 - x A[x]^3)) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[(-1)^(n - k) Binomial[n + 2 k, 3 k] Binomial[4 k, k]/(3 k + 1), {k, 0, n}], {n, 0, 24}]
  • PARI
    a(n) = sum(k=0, n, (-1)^(n-k) * binomial(n+2*k,3*k) * binomial(4*k,k) / (3*k+1)); \\ Michel Marcus, Nov 14 2021

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n+2*k,3*k) * binomial(4*k,k) / (3*k+1).
a(n) = (-1)^n*F([1/4, 1/2, 3/4, (1+n)/2, (2+n)/2, -n], [1/3, 2/3, 2/3, 1, 4/3], 2^10/3^6), where F is the generalized hypergeometric function. - Stefano Spezia, Nov 14 2021
a(n) ~ sqrt(1 - 2*r) / (4 * 2^(1/6) * sqrt(3*Pi*(1+r)) * n^(3/2) * r^(n + 1/3)), where r = 0.16809738261179529189597734361984743421471587505782537522127218... is the root of the equation 4^4 * r = 3^3 * (1+r)^3. - Vaclav Kotesovec, Nov 14 2021
From Peter Bala, Jun 02 2024: (Start)
A(x) = 1/(1 + x)*F(x/(1 + x)^3), where F(x) = Sum_{n >= 0} A002293(n)*x^n.
A(x) = 1/(1 + x) + x*A(x)^4. (End)

A349300 G.f. A(x) satisfies: A(x) = 1 / ((1 + x) * (1 - x * A(x)^4)).

Original entry on oeis.org

1, 0, 1, 4, 21, 114, 651, 3844, 23301, 144169, 906866, 5782350, 37289431, 242793439, 1593918916, 10538988984, 70121101825, 469133993094, 3154115695476, 21299373321344, 144402246424591, 982506791975780, 6706724412165956, 45917245477282994
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 13 2021

Keywords

Links

Crossrefs

Cf. A002294, A005043, A346627, A346665, A349290, A349299, A349301, A349302, A349303.

Programs

  • Mathematica
    nmax = 23; A[] = 0; Do[A[x] = 1/((1 + x) (1 - x A[x]^4)) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[(-1)^(n - k) Binomial[n + 3 k, 4 k] Binomial[5 k, k]/(4 k + 1), {k, 0, n}], {n, 0, 23}]
  • PARI
    a(n) = sum(k=0, n, (-1)^(n-k) * binomial(n+3*k,4*k) * binomial(5*k,k) / (4*k+1)); \\ Michel Marcus, Nov 14 2021

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n+3*k,4*k) * binomial(5*k,k) / (4*k+1).
a(n) = (-1)^5*F([1/5, 2/5, 3/5, 4/5, (1+n)/3, (2+n)/3, (3+n)/3, -n], [1/4, 1/2, 1/2, 3/4, 3/4, 1, 5/4], 3^3*5^5/2^16), where F is the generalized hypergeometric function. - Stefano Spezia, Nov 14 2021
a(n) ~ sqrt(1 - 3*r) / (2 * 5^(3/4) * sqrt(2*Pi*(1+r)) * n^(3/2) * r^(n + 1/4)), where r = 0.136824361675510443450981569282313811786270109272790613523286... is the root of the equation 5^5 * r = 4^4 * (1+r)^4. - Vaclav Kotesovec, Nov 14 2021
From Peter Bala, Jun 02 2024: (Start)
A(x) = 1/(1 + x)*F(x/(1 + x)^4), where F(x) = Sum_{n >= 0} A002294(n)*x^n.
A(x) = 1/(1 + x) + x*A(x)^5. (End)

A349302 G.f. A(x) satisfies: A(x) = 1 / ((1 + x) * (1 - x * A(x)^6)).

Original entry on oeis.org

1, 0, 1, 6, 43, 321, 2500, 20096, 165621, 1392397, 11896823, 103014141, 902035660, 7974080834, 71070247438, 637937825112, 5761970031357, 52329993278856, 477588786637264, 4377832437503643, 40288077072190109, 372086539388626537, 3447632819399550915
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 13 2021

Keywords

Links

Crossrefs

Cf. A002296, A005043, A346627, A346667, A349292, A349299, A349300, A349301, A349303.

Programs

  • Mathematica
    nmax = 22; A[] = 0; Do[A[x] = 1/((1 + x) (1 - x A[x]^6)) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[(-1)^(n - k) Binomial[n + 5 k, 6 k] Binomial[7 k, k]/(6 k + 1), {k, 0, n}], {n, 0, 22}]

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n+5*k,6*k) * binomial(7*k,k) / (6*k+1).
a(n) ~ sqrt(1 - 5*r) / (2 * 7^(2/3) * sqrt(3*Pi*(1+r)) * n^(3/2) * r^(n + 1/6)), where r = 0.1008057775745727124639860500770912830001828593281202101426766... is the root of the equation 7^7 * r = 6^6 * (1+r)^6. - Vaclav Kotesovec, Nov 14 2021
From Peter Bala, Jun 02 2024: (Start)
A(x) = 1/(1 + x)*F(x/(1 + x)^6), where F(x) = Sum_{n >= 0} A002296(n)*x^n.
A(x) = 1/(1 + x) + x*A(x)^7. (End)

A349303 G.f. A(x) satisfies: A(x) = 1 / ((1 + x) * (1 - x * A(x)^7)).

Original entry on oeis.org

1, 0, 1, 7, 57, 483, 4257, 38675, 359969, 3416329, 32943289, 321888455, 3180249409, 31718822793, 318934721393, 3229639622847, 32907617157641, 337144842511850, 3470986886039193, 35890957497118363, 372584381500477185, 3881595191885835547
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 13 2021

Keywords

Comments

In general, for k>=1, Sum_{j=0..n} (-1)^(n-k) * binomial(n + (k-1)*j,k*j) * binomial((k+1)*j,j) / (k*j+1) ~ sqrt(1 - (k-1)*r) / (sqrt(2*k*(k+1)*(1+r)*Pi) * (k+1)^(1/k) * n^(3/2) * r^(n + 1/k)), where r is the smallest real root of the equation (k+1)^(k+1) * r = k^k * (1+r)^k. - Vaclav Kotesovec, Nov 14 2021

Links

Crossrefs

Cf. A005043, A007556, A346627, A346668, A349293, A349299, A349300, A349301, A349302.

Programs

  • Mathematica
    nmax = 21; A[] = 0; Do[A[x] = 1/((1 + x) (1 - x A[x]^7)) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[(-1)^(n - k) Binomial[n + 6 k, 7 k] Binomial[8 k, k]/(7 k + 1), {k, 0, n}], {n, 0, 21}]

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n+6*k,7*k) * binomial(8*k,k) / (7*k+1).
a(n) ~ sqrt(1 - 6*r) / (2^(17/7) * sqrt(7*Pi*(1+r)) * n^(3/2) * r^(n + 1/7)), where r = 0.08937121041965233233945479666512758370169477786851479485467... is the real root of the equation 8^8 * r = 7^7 * (1+r)^7. - Vaclav Kotesovec, Nov 14 2021
From Peter Bala, Jun 02 2024: (Start)
A(x) = 1/(1 + x)*F(x/(1 + x)^7), where F(x) = Sum_{n >= 0} A007556(n)*x^n.
A(x) = 1/(1 + x) + x*A(x)^8. (End)
Showing 1-5 of 5 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.