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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A370042 Expansion of g.f. A(x) satisfying Sum_{n=-oo..+oo} (-1)^n * (x^n + 10*A(x))^n = 1 + 12*Sum_{n>=1} (-1)^n * x^(n^2).

Original entry on oeis.org

1, 10, 103, 1089, 11749, 128637, 1423982, 15898231, 178717112, 2020360748, 22947819571, 261696375829, 2994717484790, 34373295184712, 395580223408591, 4563146810297938, 52747346257279381, 610871638149166758, 7086520419499114527, 82334442066436896541, 957935578573905521101
Offset: 1

Views

Author

Paul D. Hanna, Feb 08 2024

Keywords

Comments

A related function is theta_4(x) = 1 + 2*Sum_{n>=1} (-1)^n * x^(n^2).

Examples

			G.f.: A(x) = x + 10*x^2 + 103*x^3 + 1089*x^4 + 11749*x^5 + 128637*x^6 + 1423982*x^7 + 15898231*x^8 + 178717112*x^9 + 2020360748*x^10 + 22947819571*x^11 + ...
where
Sum_{n=-oo..+oo} (-1)^n * (x^n + 10*A(x))^n = 1 - 12*x + 12*x^4 - 12*x^9 + 12*x^16 - 12*x^25 + 12*x^36 - 12*x^49 +- ...
SPECIAL VALUES.
(V.1) Let A = A(exp(-Pi)) = 0.07738488286995169642543180751945321776018365032150702566...
then Sum_{n=-oo..+oo} (-1)^n * (exp(-n*Pi) + 10*A)^n = 6*(Pi/2)^(1/4)/gamma(3/4) - 5 = 0.4814748289367009284...
(V.2) Let A = A(exp(-2*Pi)) = 0.001903000450057888437867399675031908155434474357834107336...
then Sum_{n=-oo..+oo} (-1)^n * (exp(-2*n*Pi) + 10*A)^n = 6*2^(1/8)*(Pi/2)^(1/4)/gamma(3/4) - 5 = 0.9775906873654428147...
(V.3) Let A = A(-exp(-Pi)) = -0.03025721520362353256298796517975081121112509387406260314...
then Sum_{n=-oo..+oo} (-1)^n * ((-1)^n*exp(-n*Pi) + 10*A)^n = 6*Pi^(1/4)/gamma(3/4) - 5 = 1.518608867279848087...
(V.4) Let A = A(-exp(-2*Pi)) = -0.001833227104147111248063467420834469150941590720555554025...
then Sum_{n=-oo..+oo} (-1)^n * ((-1)^n*exp(-2*n*Pi) + 10*A)^n = 6*sqrt(2 + sqrt(2))/2 * Pi^(1/4)/gamma(3/4) - 5 = 1.0224093129264345...

Links

Crossrefs

Cf. A370020, A370021, A370022, A370023, A370024, A370025, A370026, A370027, A370028, A370029.

Programs

  • PARI
    {a(n) = my(A=[0,1]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff( sum(m=-#A,#A, (-1)^m * (x^m + 10*Ser(A))^m ) - 1 - 12*sum(m=1,#A, (-1)^m * x^(m^2) ), #A-1)/10 ); A[n+1]}
for(n=1,30, print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) Sum_{n=-oo..+oo} (-1)^n * (x^n + 10*A(x))^n = 1 + 12*Sum_{n>=1} (-1)^n * x^(n^2).
(2) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n + 10*A(x))^(n-1) = 1 + 12*Sum_{n>=1} (-1)^n * x^(n^2).
(3) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n + 10*A(x))^n = 0.
(4) Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + 10*A(x)*x^n)^n = 1 + 12*Sum_{n>=1} (-1)^n * x^(n^2).
(5) Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + 10*A(x)*x^n)^(n+1) = 1 + 12*Sum_{n>=1} (-1)^n * x^(n^2).
(6) Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) / (1 + 10*A(x)*x^n)^(n+1) = 0.

A370040 Triangle of coefficients T(n,k) in g.f. A(x,y) satisfying Sum_{n=-oo..+oo} (-1)^n * (x^n + y*A(x,y))^n = 1 + (y+2)*Sum_{n>=1} (-1)^n * x^(n^2), for n >= 1, as read by rows.

Original entry on oeis.org

1, 0, 1, 3, 0, 1, -1, 9, 0, 1, 9, -6, 18, 0, 1, -3, 54, -19, 30, 0, 1, 22, -54, 185, -44, 45, 0, 1, -9, 264, -294, 475, -85, 63, 0, 1, 52, -324, 1463, -1026, 1020, -146, 84, 0, 1, -22, 1127, -2715, 5531, -2781, 1939, -231, 108, 0, 1, 111, -1534, 9648, -13430, 16470, -6384, 3374, -344, 135, 0, 1, -51, 4338, -19005, 51853, -49032, 41567, -13020, 5490, -489, 165, 0, 1
Offset: 1

Views

Author

Paul D. Hanna, Feb 09 2024

Keywords

Comments

A370021(n) = Sum_{k=0..n-1} T(n,k), for n >= 1.
A370022(n) = Sum_{k=0..n-1} T(n,k) * 2^k, for n >= 1.
A370023(n) = Sum_{k=0..n-1} T(n,k) * 3^k, for n >= 1.
A370024(n) = Sum_{k=0..n-1} T(n,k) * 4^k, for n >= 1.
A370025(n) = Sum_{k=0..n-1} T(n,k) * 5^k, for n >= 1.
A370026(n) = Sum_{k=0..n-1} T(n,k) * 6^k, for n >= 1.
A370027(n) = Sum_{k=0..n-1} T(n,k) * 7^k, for n >= 1.
A370028(n) = Sum_{k=0..n-1} T(n,k) * 8^k, for n >= 1.
A370029(n) = Sum_{k=0..n-1} T(n,k) * 9^k, for n >= 1.
A370042(n) = Sum_{k=0..n-1} T(n,k) * 10^k, for n >= 1.

Examples

			G.f.: A(x,y) = x*(1) + x^2*(0 + y) + x^3*(3 + y^2) + x^4*(-1 + 9*y + y^3) + x^5*(9 - 6*y + 18*y^2 + y^4) + x^6*(-3 + 54*y - 19*y^2 + 30*y^3 + y^5) + x^7*(22 - 54*y + 185*y^2 - 44*y^3 + 45*y^4 + y^6) + x^8*(-9 + 264*y - 294*y^2 + 475*y^3 - 85*y^4 + 63*y^5 + y^7) + x^9*(52 - 324*y + 1463*y^2 - 1026*y^3 + 1020*y^4 - 146*y^5 + 84*y^6 + y^8) + x^10*(-22 + 1127*y - 2715*y^2 + 5531*y^3 - 2781*y^4 + 1939*y^5 - 231*y^6 + 108*y^7 + y^9) + ...
where
Sum_{n=-oo..+oo} (-1)^n * (x^n + y*A(x,y))^n = 1 + (y+2)*Sum_{n>=1} (-1)^n * x^(n^2).
TRIANGLE.
This triangle of coefficients T(n,k) of x^n*y^k in g.f. A(x,y) begins
1;
0, 1;
3, 0, 1;
-1, 9, 0, 1;
9, -6, 18, 0, 1;
-3, 54, -19, 30, 0, 1;
22, -54, 185, -44, 45, 0, 1;
-9, 264, -294, 475, -85, 63, 0, 1;
52, -324, 1463, -1026, 1020, -146, 84, 0, 1;
-22, 1127, -2715, 5531, -2781, 1939, -231, 108, 0, 1;
111, -1534, 9648, -13430, 16470, -6384, 3374, -344, 135, 0, 1;
-51, 4338, -19005, 51853, -49032, 41567, -13020, 5490, -489, 165, 0, 1;
230, -6274, 55413, -128974, 208178, -146098, 92869, -24300, 8475, -670, 198, 0, 1; ...

Links

Crossrefs

Cf. A370020, A370021, A370022, A370023, A370024, A370025, A370026, A370027, A370028, A370029, A370042.
Cf. A370150 (column 0), A370151 (column 1), A370152 (column 2).
Cf. A370041 (dual triangle).

Programs

  • PARI
    /* Generate A(x,y) by use of definition in name */
{T(n,k) = my(A=[0,1]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff( sum(m=-#A,#A, (-1)^m * (x^m + y*Ser(A))^m ) - 1 - (y+2)*sum(m=1,#A, (-1)^m * x^(m^2) ), #A-1)/y ); H=A; polcoeff(A[n+1],k,y)}
for(n=1,15, for(k=0,n-1, print1(T(n,k),", "));print(""))
  • PARI
    /* Generate A(x,y) recursively using integration wrt y */
{T(n,k) = my(A = x +x*O(x^n), M=sqrtint(n+1), Q = sum(m=1,M, (-1)^m * x^(m^2)) +x*O(x^n));
for(i=0,n, A = (1/y) * intformal( Q / sum(m=-M,n, (-1)^m * m * (x^m + y*A)^(m-1)), y) +x*O(x^n));
polcoeff(polcoeff(A,n,x),k,y)}
for(n=1,15, for(k=0,n-1, print1(T(n,k),", "));print(""))

Formula

G.f. A(x,y) = Sum_{n>=1} Sum_{k=0..n-1} T(n,k) * x^n*y^k satisfies the following formulas.
(1) Sum_{n=-oo..+oo} (-1)^n * (x^n + y*A(x,y))^n = 1 + (y+2)*Sum_{n>=1} (-1)^n * x^(n^2).
(2) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n + y*A(x,y))^(n-1) = 1 + (y+2)*Sum_{n>=1} (-1)^n * x^(n^2).
(3) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n + y*A(x,y))^n = 0.
(4) Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + y*A(x,y)*x^n)^n = 1 + (y+2)*Sum_{n>=1} (-1)^n * x^(n^2).
(5) Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + y*A(x,y)*x^n)^(n+1) = 1 + (y+2)*Sum_{n>=1} (-1)^n * x^(n^2).
(6) Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) / (1 + y*A(x,y)*x^n)^(n+1) = 0.
(7) A(x,y) = (1/y) * Integral Q(x) / Sum_{n=-oo..+oo} (-1)^n * n * (x^n + y*A(x,y))^(n-1) dy, where Q(x) = Sum_{n>=1} (-1)^n * x^(n^2).
(8) A(x,y=0) = (1 - theta_4(x))/2 / Product_{n>=1} (1 - x^(2*n))^3, which is the g.f. of column 0 (A370150) defined at y = 0.
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