A366730 -id:A366730 - cp's OEIS frontend

A366736 Central terms of triangle A366730.

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

Offset: 0

Paul D. Hanna, Oct 30 2023

sign

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.

Paul D. Hanna, Table of n, a(n) for n = 0..50

Cf. A366730, A366731, A366732, A366733, A366734, A366735.

{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),", "))

a(n) = A366730(2*n,n) for n >= 0.

A366731 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, 0, 2, 2, 6, 19, 41, 99, 307, 750, 2062, 5776, 15674, 43700, 123729, 345728, 982580, 2801615, 7994268, 22953104, 66128105, 190846074, 552959720, 1605817449, 4673526011, 13635237816, 39860703465, 116739997283, 342538898105, 1006709394181, 2963267980415, 8735388348630

Offset: 0

Paul D. Hanna, Oct 29 2023

nonn

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

			G.f.: A(x) = 1 + x + 2*x^3 + 2*x^4 + 6*x^5 + 19*x^6 + 41*x^7 + 99*x^8 + 307*x^9 + 750*x^10 + 2062*x^11 + 5776*x^12 + 15674*x^13 + 43700*x^14 + ...

Paul D. Hanna, Table of n, a(n) for n = 0..300

Cf. A366730, A366732, A366733, A366734, A366735.

{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));A[n+1]}
for(n=0,40,print1(a(n),", "))

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) ).
a(n) ~ c * d^n / n^(3/2), where d = 3.087019811495... and c = 0.3580397646... - Vaclav Kotesovec, Jun 11 2025

A366732 Expansion of g.f. A(x) satisfying 0 = Sum_{n=-oo..+oo} x^n * A(x)^n * (2 - x^(n-1))^(n+1).

Original entry on oeis.org

1, 2, 4, 22, 108, 574, 3224, 18592, 109728, 660938, 4041900, 25034000, 156724204, 990127086, 6304425800, 40416596578, 260658078580, 1689976752116, 11008752656960, 72016455973262, 472912945955364, 3116243639293972, 20599091568973324, 136557058462319178, 907668022344460584

Offset: 0

Paul D. Hanna, Oct 29 2023

nonn

a(n) = Sum_{k=0..n} A366730(n,k) * 2^k for n >= 0.

			G.f.: A(x) = 1 + 2*x + 4*x^2 + 22*x^3 + 108*x^4 + 574*x^5 + 3224*x^6 + 18592*x^7 + 109728*x^8 + 660938*x^9 + 4041900*x^10 + 25034000*x^11 + ...

Paul D. Hanna, Table of n, a(n) for n = 0..250

Cf. A366730, A366731, A366733, A366734, A366735.

{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 * (2 - x^(n-1))^(n+1) ), #A-2));A[n+1]}
for(n=0,40,print1(a(n),", "))

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 * (2 - x^(n-1))^(n+1).
(2) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / ( A(x)^n * (1 - 2*x^(n+1))^(n-1) ).

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

Paul D. Hanna, Oct 29 2023

nonn

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

			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 + ...

Paul D. Hanna, Table of n, a(n) for n = 0..300

Cf. A366730, A366731, A366732, A366733, A366734.

{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),", "))

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

Paul D. Hanna, Oct 29 2023

nonn

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

			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 + ...

Cf. A366730, A366731, A366732, A366734, A366735.

{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),", "))

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

Paul D. Hanna, Oct 29 2023

nonn

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

			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 + ...

Cf. A366730, A366731, A366732, A366733, A366735.

{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),", "))

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) ).

cp's OEIS Frontend

A366736 Central terms of triangle A366730.

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A366731 Expansion of g.f. A(x) satisfying 0 = Sum_{n=-oo..+oo} x^n * A(x)^n * (1 - x^(n-1))^(n+1).

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A366732 Expansion of g.f. A(x) satisfying 0 = Sum_{n=-oo..+oo} x^n * A(x)^n * (2 - x^(n-1))^(n+1).

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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).

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

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

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