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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A366735 Expansion of g.f. A(x) satisfying 0 = Sum_{n=-oo..+oo} x^n * A(x)^n * (1 + (-x)^(n-1))^(n+1).

Original entry on oeis.org

1, 1, 4, 14, 54, 218, 911, 3917, 17235, 77251, 351498, 1619362, 7538944, 35412306, 167626988, 798823025, 3829325596, 18453005188, 89338777895, 434343634600, 2119679152092, 10379998771157, 50989711920778, 251194614740028, 1240735313801625, 6143268099066535
Offset: 0

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Author

Paul D. Hanna, Oct 29 2023

Keywords

Comments

a(n) = (-1)^n * Sum_{k=0..n} A366730(n,k) * (-1)^k for n >= 0.

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 14*x^3 + 54*x^4 + 218*x^5 + 911*x^6 + 3917*x^7 + 17235*x^8 + 77251*x^9 + 351498*x^10 + 1619362*x^11 + 7538944*x^12 + ...
		

Crossrefs

Programs

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

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) 0 = Sum_{n=-oo..+oo} x^n * A(x)^n * (1 + (-x)^(n-1))^(n+1).
(2) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / ( A(x)^n * (1 + (-x)^(n+1))^(n-1) ).

A366733 Expansion of g.f. A(x) satisfying 0 = Sum_{n=-oo..+oo} x^n * A(x)^n * (3 - x^(n-1))^(n+1).

Original entry on oeis.org

1, 3, 12, 90, 702, 5838, 50895, 458103, 4225683, 39745665, 379730658, 3674980518, 35951809104, 354950991006, 3532167377340, 35390917028619, 356742401734236, 3615164398809324, 36809446799831823, 376387507560832992, 3863438843523528636, 39794189982905311407
Offset: 0

Views

Author

Paul D. Hanna, Oct 29 2023

Keywords

Comments

a(n) = Sum_{k=0..n} A366730(n,k) * 3^k for n >= 0.

Examples

			G.f.: A(x) = 1 + 3*x + 12*x^2 + 90*x^3 + 702*x^4 + 5838*x^5 + 50895*x^6 + 458103*x^7 + 4225683*x^8 + 39745665*x^9 + 379730658*x^10 + ...
		

Crossrefs

Programs

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

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) 0 = Sum_{n=-oo..+oo} x^n * A(x)^n * (3 - x^(n-1))^(n+1).
(2) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / ( A(x)^n * (1 - 3*x^(n+1))^(n-1) ).

A366734 Expansion of g.f. A(x) satisfying 0 = Sum_{n=-oo..+oo} x^n * A(x)^n * (4 - x^(n-1))^(n+1).

Original entry on oeis.org

1, 4, 24, 236, 2504, 28332, 335656, 4108688, 51558000, 659737684, 8575826448, 112927383328, 1503232394344, 20195196226124, 273467339844368, 3728623506924660, 51145851271818536, 705322823588365592, 9772995790887474920, 135992755093954566300, 1899633478390401668072
Offset: 0

Views

Author

Paul D. Hanna, Oct 29 2023

Keywords

Comments

a(n) = Sum_{k=0..n} A366730(n,k) * 4^k for n >= 0.

Examples

			G.f.: A(x) = 1 + 4*x + 24*x^2 + 236*x^3 + 2504*x^4 + 28332*x^5 + 335656*x^6 + 4108688*x^7 + 51558000*x^8 + 659737684*x^9 + 8575826448*x^10 + ...
		

Crossrefs

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1,n, A = concat(A,0);
    A[#A] = polcoeff( sum(n=-#A,#A, x^n * Ser(A)^n * (4 - x^(n-1))^(n+1) ), #A-2));A[n+1]}
    for(n=0,30,print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) 0 = Sum_{n=-oo..+oo} x^n * A(x)^n * (4 - x^(n-1))^(n+1).
(2) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / ( A(x)^n * (1 - 4*x^(n+1))^(n-1) ).

A366736 Central terms of triangle A366730.

Original entry on oeis.org

1, -2, 14, -176, 2615, -42444, 734310, -13332898, 251087228, -4863520344, 96340129818, -1943639738074, 39815238143374, -826201916477272, 17334983283537509, -367213838120451038, 7844257467257818627, -168807941163188191336, 3656662240133060807499, -79675906058698383705100
Offset: 0

Views

Author

Paul D. Hanna, Oct 30 2023

Keywords

Comments

This sequence is defined by a(n) = [x^(2*n)*y^n] F(x,y) for n >= 0, where F(x,y) satisfies 0 = Sum_{n=-oo..+oo} x^n * F(x,y)^n * (y - x^(n-1))^(n+1), and F(x,y) is the g.f. of triangle A366730.

Crossrefs

Programs

  • PARI
    {A366730(n,k) = my(A=[1]); for(i=1,n, A = concat(A,0);
    A[#A] = polcoeff( sum(n=-#A,#A, x^n * Ser(A)^n * (y - x^(n-1))^(n+1) ), #A-2)); polcoeff(A[n+1],k)}
    for(n=0,20, print1(A366730(2*n,n),", "))

Formula

a(n) = A366730(2*n,n) for n >= 0.
Previous Showing 11-14 of 14 results.