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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A354656 Column 3 of triangle A354650: a(n) = A354650(n,3), for n >= 1.

Original entry on oeis.org

1, 30, 340, 2530, 14595, 70737, 301070, 1157820, 4100785, 13563010, 42321840, 125586440, 356621070, 973989030, 2569116330, 6567458520, 16317741975, 39504992395, 93390535840, 215983566780, 489454806785, 1088433416785, 2378160809610, 5111208572940, 10816601842950
Offset: 1

Views

Author

Paul D. Hanna, Jun 02 2022

Keywords

Links

Crossrefs

Cf. A354649, A354650, A354655, A354657.

Programs

  • PARI
    {A354650(n,k) = my(A=[1+y]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff(y + sum(m=0,sqrtint(2*#A+9), (-1)^m * x^(m*(m-1)/2) * (1 - x^(2*m+1)) * Ser(A)^(m*(m+1)/2) ),#A-1) );
polcoeff(A[n+1],k,y)}
for(n=1,30,print1(A354650(n,3),", "))

Formula

a(n) = (-1)^n * A354649(n,3), for n >= 1.
a(n) = A354650(n,3), for n >= 1.

A354659 A diagonal of triangle A354650: a(n) = A354650(n,n+1), for n >= 0.

Original entry on oeis.org

1, 3, 30, 390, 5928, 98910, 1757688, 32683680, 628884300, 12428334215, 250940544738, 5156722096422, 107538413657010, 2270751678647100, 48464836803383400, 1044050265679857144, 22675350105240015204, 496034970650911331550, 10920742396832034391590
Offset: 0

Views

Author

Paul D. Hanna, Jun 02 2022

Keywords

Links

Crossrefs

Cf. A354649, A354650, A354658, A354660.

Programs

  • PARI
    {A354650(n,k) = my(A=[1+y]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff(y + sum(m=0,sqrtint(2*#A+9), (-1)^m * x^(m*(m-1)/2) * (1 - x^(2*m+1)) * Ser(A)^(m*(m+1)/2) ),#A-1) );
polcoeff(A[n+1],k,y)}
for(n=0,20,print1(A354650(n,n+1),", "))

Formula

a(n) = A354649(n,n+1), for n >= 0.
a(n) = A354650(n,n+1), for n >= 0.
a(n) ~ c * d^n / n^2, where d = 24.5759928778699928131449756... and c = 0.35661791857107638456206... - Vaclav Kotesovec, Mar 19 2023

A354660 a(n) = A354650(n,2*n), for n >= 0.

Original entry on oeis.org

1, 3, 15, 83, 486, 2937, 18109, 113220, 715122, 4552229, 29156985, 187683795, 1213110600, 7868238588, 51184173036, 333809308696, 2181842704602, 14288748463485, 93737673347185, 615889045662345, 4052198020223430, 26694405836621985, 176052003674681925
Offset: 0

Views

Author

Paul D. Hanna, Jun 02 2022

Keywords

Links

Crossrefs

Cf. A354649, A354650, A354658, A354659.

Programs

  • PARI
    {A354650(n,k) = my(A=[1+y]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff(y + sum(m=0,sqrtint(2*#A+9), (-1)^m * x^(m*(m-1)/2) * (1 - x^(2*m+1)) * Ser(A)^(m*(m+1)/2) ),#A-1) );
polcoeff(A[n+1],k,y)}
for(n=0,20,print1(A354650(n,2*n),", "))

Formula

a(n) = (-1)^(n+1) * A354649(n,2*n), for n >= 0.
a(n) = A354650(n,2*n), for n >= 0.
a(n) ~ 13 * 3^(3*n - 5/2) / (sqrt(Pi*n) * 2^(2*n)). - Vaclav Kotesovec, Mar 19 2023

A354657 a(n) = A354655(n)/3, for n >= 1.

Original entry on oeis.org

1, 9, 49, 210, 765, 2492, 7434, 20700, 54420, 136360, 327789, 760102, 1707342, 3728025, 7935525, 16507152, 33624045, 67186077, 131891825, 254710260, 484474753, 908538081, 1681364124, 3073166600, 5551851375, 9919925145, 17541289017, 30714092066, 53279031420
Offset: 1

Views

Author

Paul D. Hanna, Jun 02 2022

Keywords

Links

Crossrefs

Cf. A354649, A354650, A354655.

Programs

  • PARI
    {A354650(n,k) = my(A=[1+y]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff(y + sum(m=0,sqrtint(2*#A+9), (-1)^m * x^(m*(m-1)/2) * (1 - x^(2*m+1)) * Ser(A)^(m*(m+1)/2) ),#A-1) );
polcoeff(A[n+1],k,y)}
for(n=1,30,print1(A354650(n,2)/3,", "))

Formula

a(n) = A354655(n)/3, for n >= 1.
a(n) = A354650(n,2)/3, for n >= 1.
a(n) = (-1)^(n+1) * A354649(n,2)/3, for n >= 1.

A354647 G.f. A(x) satisfies: -x^2 = Sum_{n=-oo..oo} (-1)^n * x^(n*(n+1)/2) * A(x)^(n*(n-1)/2).

Original entry on oeis.org

1, 0, 1, 3, 9, 25, 78, 256, 881, 3064, 10831, 38766, 140550, 514625, 1900301, 7067013, 26448613, 99539716, 376489459, 1430330451, 5455742957, 20885223619, 80213926069, 309002022843, 1193616950854, 4622372591972, 17942238661229, 69795082381496, 272046051362013
Offset: 0

Views

Author

Paul D. Hanna, Jun 21 2022

Keywords

Examples

			G.f.: A(x) = 1 + x^2 + 3*x^3 + 9*x^4 + 25*x^5 + 78*x^6 + 256*x^7 + 881*x^8 + 3064*x^9 + 10831*x^10 + 38766*x^11 + 140550*x^12 + ...
such that A = A(x) satisfies:
(1) -x^2 = ... + x^36*A^28 - x^28*A^21 + x^21*A^15 - x^15*A^10 + x^10*A^6 - x^6*A^3 + x^3*A - x + 1 - A + x*A^3 - x^3*A^6 + x^6*A^10 - x^10*A^15 + x^15*A^21 - x^21*A^28 + x^28*A^36 + ...
(2) -x^2 = (1-x) - (1-x^3)*A + x*(1-x^5)*A^3 - x^3*(1-x^7)*A^6 + x^6*(1-x^9)*A^10 - x^10*(1-x^11)*A^15 + x^15*(1-x^13)*A^21 - x^21*(1-x^15)*A^28 + ...
(3) -x^2 = (1-A) - (1-A^3)*x + A*(1-A^5)*x^3 - A^3*(1-A^7)*x^6 + A^6*(1-A^9)*x^10 - A^10*(1-A^11)*x^15 + A^15*(1-A^13)*x^21 - A^21*(1-A^15)*x^28 + ...
(4) -x^2 = (1 - x*A)*(1 - A)*(1-x) * (1 - x^2*A^2)*(1 - x*A^2)*(1 - x^2*A) * (1 - x^3*A^3)*(1 - x^2*A^3)*(1 - x^3*A^2) * (1 - x^4*A^4)*(1 - x^3*A^4)*(1 - x^4*A^3) * (1 - x^5*A^5)*(1 - x^4*A^5)*(1 - x^5*A^4) * ...

Crossrefs

Cf. A268650, A354648, A354649.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff(x^2 + sum(m=0,sqrtint(2*#A+9), (-1)^m * x^(m*(m-1)/2) * (1 - x^(2*m+1)) * Ser(A)^(m*(m+1)/2) ),#A-1) );A[n+1]}
for(n=0,30,print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
(1) -x^2 = Sum_{n=-oo..oo} (-1)^n * x^(n*(n-1)/2) * A(x)^(n*(n+1)/2).
(2) -x^2 = Sum_{n>=0} (-1)^n * x^(n*(n-1)/2) * (1 - x^(2*n+1)) * A(x)^(n*(n+1)/2).
(3) -x^2 = Sum_{n>=0} (-1)^n * A(x)^(n*(n-1)/2) * (1 - A(x)^(2*n+1)) * x^(n*(n+1)/2).
(4) -x^2 = Product_{n>=1} (1 - x^n*A(x)^n) * (1 - x^(n-1)*A(x)^n) * (1 - x^n*A(x)^(n-1)), by the Jacobi triple product identity.

A354648 G.f. A(x) satisfies: -x^3 = Sum_{n=-oo..oo} (-1)^n * x^(n*(n+1)/2) * A(x)^(n*(n-1)/2).

Original entry on oeis.org

1, 0, 0, 1, 3, 9, 22, 54, 135, 368, 1060, 3135, 9295, 27472, 81309, 242255, 728429, 2208483, 6736523, 20634196, 63410076, 195467757, 604457802, 1875053982, 5833449236, 18195767301, 56888745654, 178238369769, 559538565187, 1759796017533, 5544359742297
Offset: 0

Views

Author

Paul D. Hanna, Jun 21 2022

Keywords

Examples

			G.f.: A(x) = 1 + x^3 + 3*x^4 + 9*x^5 + 22*x^6 + 54*x^7 + 135*x^8 + 368*x^9 + 1060*x^10 + 3135*x^11 + 9295*x^12 + 27472*x^13 + ...
such that A = A(x) satisfies:
(1) -x^3 = ... + x^36*A^28 - x^28*A^21 + x^21*A^15 - x^15*A^10 + x^10*A^6 - x^6*A^3 + x^3*A - x + 1 - A + x*A^3 - x^3*A^6 + x^6*A^10 - x^10*A^15 + x^15*A^21 - x^21*A^28 + x^28*A^36 + ...
(2) -x^3 = (1-x) - (1-x^3)*A + x*(1-x^5)*A^3 - x^3*(1-x^7)*A^6 + x^6*(1-x^9)*A^10 - x^10*(1-x^11)*A^15 + x^15*(1-x^13)*A^21 - x^21*(1-x^15)*A^28 + ...
(3) -x^3 = (1-A) - (1-A^3)*x + A*(1-A^5)*x^3 - A^3*(1-A^7)*x^6 + A^6*(1-A^9)*x^10 - A^10*(1-A^11)*x^15 + A^15*(1-A^13)*x^21 - A^21*(1-A^15)*x^28 + ...
(4) -x^3 = (1 - x*A)*(1 - A)*(1-x) * (1 - x^2*A^2)*(1 - x*A^2)*(1 - x^2*A) * (1 - x^3*A^3)*(1 - x^2*A^3)*(1 - x^3*A^2) * (1 - x^4*A^4)*(1 - x^3*A^4)*(1 - x^4*A^3) * (1 - x^5*A^5)*(1 - x^4*A^5)*(1 - x^5*A^4) * ...

Crossrefs

Cf. A268650, A354647, A354649, A000716.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff(x^3 + sum(m=0,sqrtint(2*#A+9), (-1)^m * x^(m*(m-1)/2) * (1 - x^(2*m+1)) * Ser(A)^(m*(m+1)/2) ),#A-1) );A[n+1]}
for(n=0,30,print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
(1) -x^3 = Sum_{n=-oo..oo} (-1)^n * x^(n*(n-1)/2) * A(x)^(n*(n+1)/2).
(2) -x^3 = Sum_{n>=0} (-1)^n * x^(n*(n-1)/2) * (1 - x^(2*n+1)) * A(x)^(n*(n+1)/2).
(3) -x^3 = Sum_{n>=0} (-1)^n * A(x)^(n*(n-1)/2) * (1 - A(x)^(2*n+1)) * x^(n*(n+1)/2).
(4) -x^3 = Product_{n>=1} (1 - x^n*A(x)^n) * (1 - x^(n-1)*A(x)^n) * (1 - x^n*A(x)^(n-1)), by the Jacobi triple product identity.
Previous Showing 11-16 of 16 results.

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