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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A370041 Triangle of coefficients T(n,k) in g.f. A(x,y) satisfying Sum_{n=-oo..+oo} (x^n - y*A(x,y))^n = 1 - (y-2)*Sum_{n>=1} x^(n^2), for n >= 1, as read by rows.

Original entry on oeis.org

1, 0, 1, -1, 0, 1, 1, -3, 0, 1, 1, 6, -6, 0, 1, -1, 6, 19, -10, 0, 1, -2, -18, 17, 44, -15, 0, 1, 1, -4, -98, 35, 85, -21, 0, 1, 4, 36, 39, -334, 60, 146, -28, 0, 1, -2, 11, 291, 311, -879, 91, 231, -36, 0, 1, -5, -74, -264, 1310, 1286, -1960, 126, 344, -45, 0, 1, 3, -30, -627, -2547, 4248, 3935, -3892, 162, 489, -55, 0, 1
Offset: 1

Views

Author

Paul D. Hanna, Feb 10 2024

Keywords

Comments

A370031(n) = Sum_{k=0..n-1} T(n,k), for n >= 1.
A355868(n) = Sum_{k=0..n-1} T(n,k) * 2^k, for n >= 1.
A370033(n) = Sum_{k=0..n-1} T(n,k) * 3^k, for n >= 1.
A370034(n) = Sum_{k=0..n-1} T(n,k) * 4^k, for n >= 1.
A370035(n) = Sum_{k=0..n-1} T(n,k) * 5^k, for n >= 1.
A370036(n) = Sum_{k=0..n-1} T(n,k) * 6^k, for n >= 1.
A370037(n) = Sum_{k=0..n-1} T(n,k) * 7^k, for n >= 1.
A370038(n) = Sum_{k=0..n-1} T(n,k) * 8^k, for n >= 1.
A370039(n) = Sum_{k=0..n-1} T(n,k) * 9^k, for n >= 1.
A370043(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*(-1 + y^2) + x^4*(1 - 3*y + y^3) + x^5*(1 + 6*y - 6*y^2 + y^4) + x^6*(-1 + 6*y + 19*y^2 - 10*y^3 + y^5) + x^7*(-2 - 18*y + 17*y^2 + 44*y^3 - 15*y^4 + y^6) + x^8*(1 - 4*y - 98*y^2 + 35*y^3 + 85*y^4 - 21*y^5 + y^7) + x^9*(4 + 36*y + 39*y^2 - 334*y^3 + 60*y^4 + 146*y^5 - 28*y^6 + y^8) + x^10*(-2 + 11*y + 291*y^2 + 311*y^3 - 879*y^4 + 91*y^5 + 231*y^6 - 36*y^7 + y^9) + ...
where
Sum_{n=-oo..+oo} (x^n - y*A(x,y))^n = 1 - (y-2)*Sum_{n>=1} 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;
  -1, 0, 1;
  1, -3, 0, 1;
  1, 6, -6, 0, 1;
  -1, 6, 19, -10, 0, 1;
  -2, -18, 17, 44, -15, 0, 1;
  1, -4, -98, 35, 85, -21, 0, 1;
  4, 36, 39, -334, 60, 146, -28, 0, 1;
  -2, 11, 291, 311, -879, 91, 231, -36, 0, 1;
  -5, -74, -264, 1310, 1286, -1960, 126, 344, -45, 0, 1;
  3, -30, -627, -2547, 4248, 3935, -3892, 162, 489, -55, 0, 1;
  6, 178, 773, -2626, -12982, 11138, 9989, -7092, 195, 670, -66, 0, 1;
  -4, 40, 1525, 10094, -5842, -48126, 25138, 22258, -12093, 220, 891, -78, 0, 1;
  ...

Links

Crossrefs

Cf. A370030, A370031, A355868, A370033, A370034, A370035, A370036, A370037, A370038, A370039, A370043.
Cf. A370153 (column 0), A370154 (column 1), A370155 (column 2).
Cf. A370040 (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=-sqrtint(#A+1),#A, (x^m - y*Ser(A))^m ) - 1 + (y-2)*sum(m=1,sqrtint(#A+1), x^(m^2) ), #A-1)/y ); 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, x^(m^2)) +x*O(x^n));
for(i=0, n, A = (1/y) * intformal( Q / sum(m=-M, n, 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} T(n,k)*x^n*y^k satisfies the following formulas.
(1) Sum_{n=-oo..+oo} (x^n - y*A(x,y))^n = 1 - (y-2)*Sum_{n>=1} x^(n^2).
(2) Sum_{n=-oo..+oo} x^n * (x^n + y*A(x,y))^(n-1) = 1 - (y-2)*Sum_{n>=1} 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} x^(n^2) / (1 - x^n*y*A(x,y))^n = 1 - (y-2)*Sum_{n>=1} x^(n^2).
(5) Sum_{n=-oo..+oo} x^(n^2) / (1 + x^n*y*A(x,y))^(n+1) = 1 - (y-2)*Sum_{n>=1} x^(n^2).
(6) Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 - x^n*y*A(x,y))^n = 0.
(7) A(x,y) = (1/y) * Integral Q(x) / Sum_{n=-oo..+oo} n * (x^n - y*A(x,y))^(n-1) dy, where Q(x) = Sum_{n>=1} x^(n^2).
(8) A(x,y=0) = (theta_3(x) - 1)/2 * Product_{n>=1} (1 - x^(4*n-2)) / (1 - x^(4*n)), which is the g.f. of column 0 (A370153) defined at y = 0.

A357227 a(n) = coefficient of x^n, n >= 0, in A(x) such that: 1 = Sum_{n=-oo..+oo} x^n * (2*A(x) - x^n)^(n-1).

Original entry on oeis.org

1, 1, 5, 27, 156, 961, 6145, 40546, 273784, 1883468, 13153544, 93012247, 664640794, 4791939802, 34816034143, 254659426691, 1873698891024, 13858201221637, 102975937795619, 768385165594607, 5755185884844403, 43253819566052165, 326093530416255178, 2465456045342545908
Offset: 0

Views

Author

Paul D. Hanna, Oct 17 2022

Keywords

Comments

Related identity: 0 = Sum_{n=-oo..+oo} x^n * (y - x^n)^n, which holds formally for all y.

Examples

			G.f.: A(x) = 1 + x + 5*x^2 + 27*x^3 + 156*x^4 + 961*x^5 + 6145*x^6 + 40546*x^7 + 273784*x^8 + 1883468*x^9 + 13153544*x^10 + 93012247*x^11 + 664640794*x^12 + ...
where
1 = ... + x^(-3)/(2*A(x) - x^(-3))^4 + x^(-2)/(2*A(x) - x^(-2))^3 + x^(-1)/(2*A(x) - x^(-1))^2 + 1/(2*A(x) - 1) + x + x^2*(2*A(x) - x^2) + x^3*(2*A(x) - x^3)^2 + x^4*(2*A(x) - x^4)^3 + ... + x^n*(2*A(x) - x^n)^(n-1) + ...
SPECIFIC VALUES.
A(1/9) = 1.30108724398914093656591796643458817060949...
A(1/10) = 1.22176622612326449515553495048940456186175...

Links

Crossrefs

Cf. A358961, A358962, A358963, A358964, A358965, A358937.
Cf. A355868, A357232, A355865.
Cf. A363312, A363313, A363314, A363315.

Programs

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

Formula

Generating function A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) 1 = Sum_{n=-oo..+oo} x^n * (2*A(x) - x^n)^(n-1).
(2) 2*A(x) = Sum_{n=-oo..+oo} x^(2*n) * (2*A(x) - x^n)^(n-1).
(3) 1 = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(n^2) / (1 - 2*x^n*A(x))^(n+1).
(4) 2*A(x) = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(n*(n-1)) / (1 - 2*x^n*A(x))^(n+1).

A370030 Table in which the g.f. of row n, R(n,x), satisfies Sum_{k=-oo..+oo} (x^k - n*R(n,x))^k = 1 - (n-2)*Sum_{k>=1} x^(k^2), for n >= 1, as read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 0, 1, 3, 3, -1, 1, 4, 8, 3, 2, 1, 5, 15, 19, 5, 15, 1, 6, 24, 53, 46, 39, 27, 1, 7, 35, 111, 185, 161, 206, -1, 1, 8, 48, 199, 506, 711, 799, 697, -76, 1, 9, 63, 323, 1117, 2379, 3270, 4021, 1656, 19, 1, 10, 80, 489, 2150, 6335, 12083, 17297, 17932, 3208, 719, 1, 11, 99, 703, 3761, 14349, 37222, 67531, 95108, 71311, 8727, 1687
Offset: 1

Views

Author

Paul D. Hanna, Feb 10 2024

Keywords

Comments

A related function is theta_3(x) = 1 + 2*Sum_{n>=1} x^(n^2).

Examples

			This table of coefficients T(n,k) of x^k in R(n,x), n >= 1, k >= 1, begins:
A370031: [1,  1,  0,  -1,    2,    15,     27,      -1,      -76, ...];
A355868: [1,  2,  3,   3,    5,    39,    206,     697,     1656, ...];
A370033: [1,  3,  8,  19,   46,   161,    799,    4021,    17932, ...];
A370034: [1,  4, 15,  53,  185,   711,   3270,   17297,    95108, ...];
A370035: [1,  5, 24, 111,  506,  2379,  12083,   67531,   406284, ...];
A370036: [1,  6, 35, 199, 1117,  6335,  37222,  230809,  1515784, ...];
A370037: [1,  7, 48, 323, 2150, 14349,  97431,  681857,  4956116, ...];
A370038: [1,  8, 63, 489, 3761, 28911, 224174, 1768801, 14298852, ...];
A370039: [1,  9, 80, 703, 6130, 53351, 466315, 4118167, 36941188, ...];
A370043: [1, 10, 99, 971, 9461, 91959, 895518, 8775161, 86870264, ...]; ...
...
where the n-th row function R(n,x) satisfies
Sum_{k=-oo..+oo} (x^k - n*R(n,x))^k = 1 - (n-2)*Sum_{k>=1} x^(k^2).

Links

Crossrefs

Cf. A370031, A355868, A370033, A370034, A370035, A370036, A370037, A370038, A370039, A370043.
Cf. A370041, A370020 (dual table).

Programs

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

Formula

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

A370031 Expansion of g.f. A(x) satisfying Sum_{n=-oo..+oo} (x^n - A(x))^n = Sum_{n>=0} x^(n^2).

Original entry on oeis.org

1, 1, 0, -1, 2, 15, 27, -1, -76, 19, 719, 1687, 184, -5976, -3749, 44093, 130933, 42026, -512833, -667101, 2976177, 11391169, 6608432, -45604863, -87819235, 202544340, 1053407806, 922859161, -4085924365, -10600384406, 12656739909, 100646660458, 121472828448, -360976456530
Offset: 1

Views

Author

Paul D. Hanna, Feb 10 2024

Keywords

Comments

A related function is theta_3(x) = 1 + 2*Sum_{n>=1} x^(n^2).

Examples

			G.f.: A(x) = 1 + x - x^3 + 2*x^4 + 15*x^5 + 27*x^6 - x^7 - 76*x^8 + 19*x^9 + 719*x^10 + 1687*x^11 + 184*x^12 - 5976*x^13 - 3749*x^14 + 44093*x^15 + ...
where
Sum_{n=-oo..+oo} (x^n - A(x))^n = 1 + x + x^4 + x^9 + x^16 + x^25 + x^36 + x^49 + ...
SPECIAL VALUES.
(V.1) Let A = A(exp(-Pi)) = 0.04507828029039130528308497098432879536368681539259286273...
then Sum_{n=-oo..+oo} (exp(-n*Pi) - A)^n = (1 + Pi^(1/4)/gamma(3/4))/2 = 1.0432174056066540...
(V.2) Let A = A(exp(-2*Pi)) = 0.001870930061948701432816606547007172908053584772650237678...
then Sum_{n=-oo..+oo} (exp(-2*n*Pi) - A)^n = (1 + sqrt(2 + sqrt(2))/2 * Pi^(1/4)/gamma(3/4))/2 = 1.00186744274386954...
(V.3) Let A = A(-exp(-Pi)) = -0.04135017416264159536574596265267969182735801577042441264...
then Sum_{n=-oo..+oo} ((-1)^n*exp(-n*Pi) - A)^n = (1 + (Pi/2)^(1/4)/gamma(3/4))/2 = 0.9567895690780584107...
(V.4) Let A = A(-exp(-2*Pi)) = -0.001863955401558124515592555303127910405358304631205735085...
then Sum_{n=-oo..+oo} ((-1)^n*exp(-2*n*Pi) - A)^n = (1 + 2^(1/8)*(Pi/2)^(1/4)/gamma(3/4))/2 = 0.99813255728045356...

Links

Crossrefs

Cf. A370041, A370030, A355868, A370033, A370034, A370035, A370036, A370037, A370038, A370039, A370043.

Programs

  • PARI
    {a(n) = my(A=[0,1]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff( sum(n=-#A,#A, (x^n - Ser(A))^n ) - sum(n=0,#A, x^(n^2) ), #A-1) ); A[n+1]}
for(n=1,40, print1(a(n),", "))

Formula

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

A370033 Expansion of g.f. A(x) satisfying Sum_{n=-oo..+oo} (x^n - 3*A(x))^n = 1 - Sum_{n>=1} x^(n^2).

Original entry on oeis.org

1, 3, 8, 19, 46, 161, 799, 4021, 17932, 71311, 268639, 1045731, 4464576, 20500010, 95221503, 429913365, 1879365529, 8112744634, 35452835755, 158833086233, 725458442577, 3329609464605, 15194309369384, 68837584452055, 311257278509193, 1413730859134250, 6469321177004978
Offset: 1

Views

Author

Paul D. Hanna, Feb 10 2024

Keywords

Comments

A related function is theta_3(x) = 1 + 2*Sum_{n>=1} x^(n^2).

Examples

			G.f.: A(x) = 1 + 3*x + 8*x^2 + 19*x^3 + 46*x^4 + 161*x^5 + 799*x^6 + 4021*x^7 + 17932*x^8 + 71311*x^9 + 268639*x^10 + 1045731*x^11 + 4464576*x^12 + ...
where
Sum_{n=-oo..+oo} (x^n - 3*A(x))^n = 1 - x - x^4 - x^9 - x^16 - x^25 - x^36 - x^49 - ...
SPECIAL VALUES.
(V.1) Let A = A(exp(-Pi)) = 0.04953636800560980886288845724196786482586224709976648461...
then Sum_{n=-oo..+oo} (exp(-n*Pi) - 3*A)^n = (3 - Pi^(1/4)/gamma(3/4))/2 = 0.956782594393345992...
(V.2) Let A = A(exp(-2*Pi)) = 0.001877957090194880545086201853719041435355287864597005509...
then Sum_{n=-oo..+oo} (exp(-2*n*Pi) - 3*A)^n = (3 - sqrt(2 + sqrt(2))/2 * Pi^(1/4)/gamma(3/4))/2 = 0.998132557256130454...
(V.3) Let A = A(-exp(-Pi)) = -0.03819699447470815952471171970837842342724818247967540335...
then Sum_{n=-oo..+oo} ((-1)^n*exp(-n*Pi) - 3*A)^n = (3 - (Pi/2)^(1/4)/gamma(3/4))/2 = 1.0432104309219415...
(V.4) Let A = A(-exp(-2*Pi)) = -0.001857032573904813918259314464039219802478066024973444789...
then Sum_{n=-oo..+oo} ((-1)^n*exp(-2*n*Pi) - 3*A)^n = (3 - 2^(1/8)*(Pi/2)^(1/4)/gamma(3/4))/2 = 1.001867442719546432...

Links

Crossrefs

Cf. A370041, A370030, A370031, A355868, A370034, A370035, A370036, A370037, A370038, A370039, A370043.

Programs

  • PARI
    {a(n) = my(A=[0,1]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff( sum(m=-#A,#A, (x^m - 3*Ser(A))^m ) - 1 + sum(m=1,#A, x^(m^2) ), #A-1)/3 ); 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} (x^n - 3*A(x))^n = 1 - Sum_{n>=1} x^(n^2).
(2) Sum_{n=-oo..+oo} x^n * (x^n + 3*A(x))^(n-1) = 1 - Sum_{n>=1} x^(n^2).
(3) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n - 3*A(x))^n = 0.
(4) Sum_{n=-oo..+oo} x^(n^2) / (1 - 3*x^n*A(x))^n = 1 - Sum_{n>=1} x^(n^2).
(5) Sum_{n=-oo..+oo} x^(n^2) / (1 + 3*x^n*A(x))^(n+1) = 1 - Sum_{n>=1} x^(n^2).
(6) Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 - 3*x^n*A(x))^n = 0.

A370034 Expansion of g.f. A(x) satisfying Sum_{n=-oo..+oo} (x^n - 4*A(x))^n = 1 - 2*Sum_{n>=1} x^(n^2).

Original entry on oeis.org

1, 4, 15, 53, 185, 711, 3270, 17297, 95108, 511258, 2653139, 13479835, 68633758, 356913516, 1906525759, 10388550830, 57084621325, 313692565172, 1719365476703, 9416232699651, 51699722653269, 285294478988749, 1583233662850172, 8826549215612727, 49354550054780111, 276444281747417079
Offset: 1

Views

Author

Paul D. Hanna, Feb 10 2024

Keywords

Comments

A related function is theta_3(x) = 1 + 2*Sum_{n>=1} x^(n^2).

Examples

			G.f.: A(x) = x + 4*x^2 + 15*x^3 + 53*x^4 + 185*x^5 + 711*x^6 + 3270*x^7 + 17297*x^8 + 95108*x^9 + 511258*x^10 + 2653139*x^11 + 13479835*x^12 + ...
where
Sum_{n=-oo..+oo} (x^n - 4*A(x))^n = 1 - 2*x - 2*x^4 - 2*x^9 - 2*x^16 - 2*x^25 - 2*x^36 - 2*x^49 - ...
SPECIAL VALUES.
(V.1) Let A = A(exp(-Pi)) = 0.05211271680112049721451382589099198923178830298930738503...
then Sum_{n=-oo..+oo} (exp(-n*Pi) - 4*A)^n = 2 - Pi^(1/4)/gamma(3/4) = 0.913565188786691985...
(V.2) Let A = A(exp(-2*Pi)) = 0.001881490436109324727231096204943046774873234177072692211...
then Sum_{n=-oo..+oo} (exp(-2*n*Pi) - 4*A)^n = 2 - sqrt(2 + sqrt(2))/2 * Pi^(1/4)/gamma(3/4) = 0.99626511451226...
(V.3) Let A = A(-exp(-Pi)) = -0.03679381086518350821622244996144281973183248006035375080...
then Sum_{n=-oo..+oo} ((-1)^n*exp(-n*Pi) - 4*A)^n = 2 - (Pi/2)^(1/4)/gamma(3/4) = 1.08642086184388317...
(V.4) Let A = A(-exp(-2*Pi)) = -0.001853590408074327278987912837104527635895010708605840824...
then Sum_{n=-oo..+oo} ((-1)^n*exp(-2*n*Pi) - 4*A)^n = 2 - 2^(1/8)*(Pi/2)^(1/4)/gamma(3/4) = 1.003734885439...

Links

Crossrefs

Cf. A370041, A370030, A370031, A355868, A370033, A370035, A370036, A370037, A370038, A370039, A370043.
Cf. A369671.

Programs

  • PARI
    {a(n) = my(A=[0,1]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff( sum(m=-#A,#A, (x^m - 4*Ser(A))^m ) - 1 + 2*sum(m=1,#A, x^(m^2) ), #A-1)/4 ); 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} (x^n - 4*A(x))^n = 1 - 2*Sum_{n>=1} x^(n^2).
(2) Sum_{n=-oo..+oo} x^n * (x^n + 4*A(x))^(n-1) = 1 - 2*Sum_{n>=1} x^(n^2).
(3) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n - 4*A(x))^n = 0.
(4) Sum_{n=-oo..+oo} x^(n^2) / (1 - 4*x^n*A(x))^n = 1 - 2*Sum_{n>=1} x^(n^2).
(5) Sum_{n=-oo..+oo} x^(n^2) / (1 + 4*x^n*A(x))^(n+1) = 1 - 2*Sum_{n>=1} x^(n^2).
(6) Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 - 4*x^n*A(x))^n = 0.

A370035 Expansion of g.f. A(x) satisfying Sum_{n=-oo..+oo} (x^n - 5*A(x))^n = 1 - 3*Sum_{n>=1} x^(n^2).

Original entry on oeis.org

1, 5, 24, 111, 506, 2379, 12083, 67531, 406284, 2531203, 15866775, 98883303, 612775096, 3798083196, 23698615411, 149450139421, 953022469813, 6132672546362, 39706366904663, 258032916789711, 1680779512045521, 10970344718827785, 71764325720800072, 470691291168007709
Offset: 1

Views

Author

Paul D. Hanna, Feb 10 2024

Keywords

Comments

A related function is theta_3(x) = 1 + 2*Sum_{n>=1} x^(n^2).

Examples

			G.f.: A(x) = x + 5*x^2 + 24*x^3 + 111*x^4 + 506*x^5 + 2379*x^6 + 12083*x^7 + 67531*x^8 + 406284*x^9 + 2531203*x^10 + 15866775*x^11 + 98883303*x^12 + ...
where
Sum_{n=-oo..+oo} (x^n - 5*A(x))^n = 1 - 3*x - 3*x^4 - 3*x^9 - 3*x^16 - 3*x^25 - 3*x^36 - 3*x^49 - ...
\SPECIAL VALUES.
(V.1) Let A = A(exp(-Pi)) = 0.05497127752043377386868704294930896868077597772598908285...
then Sum_{n=-oo..+oo} (exp(-n*Pi) - 5*A)^n = (5 - 3*Pi^(1/4)/gamma(3/4) )/2 = 0.870347783180037978...
(V.2) Let A = A(exp(-2*Pi)) = 0.001885037102906729934432374294398706956703235597857256076...
then Sum_{n=-oo..+oo} (exp(-2*n*Pi) - 5*A)^n = (5 - 3*sqrt(2 + sqrt(2))/2 * Pi^(1/4)/gamma(3/4))/2 = 0.994397671768391...
(V.3) Let A = A(-exp(-Pi)) = -0.03548990756971248576955893224372969073755967165800772531...
then Sum_{n=-oo..+oo} ((-1)^n*exp(-n*Pi) - 5*A)^n = (5 - 3*(Pi/2)^(1/4)/gamma(3/4))/2 = 1.12963129276582476...
(V.4) Let A = A(-exp(-2*Pi)) = -0.001850160979277236538611428135062916090397865766804127684...
then Sum_{n=-oo..+oo} ((-1)^n*exp(-2*n*Pi) - 5*A)^n = (5 - 3*2^(1/8)*(Pi/2)^(1/4)/gamma(3/4))/2 = 1.005602328158639...

Links

Crossrefs

Cf. A370041, A370030, A370031, A355868, A370033, A370034, A370036, A370037, A370038, A370039, A370043.

Programs

  • PARI
    {a(n) = my(A=[0,1]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff( sum(m=-#A,#A, (x^m - 5*Ser(A))^m ) - 1 + 3*sum(m=1,#A, x^(m^2) ), #A-1)/5 ); 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} (x^n - 5*A(x))^n = 1 - 3*Sum_{n>=1} x^(n^2).
(2) Sum_{n=-oo..+oo} x^n * (x^n + 5*A(x))^(n-1) = 1 - 3*Sum_{n>=1} x^(n^2).
(3) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n - 5*A(x))^n = 0.
(4) Sum_{n=-oo..+oo} x^(n^2) / (1 - 5*x^n*A(x))^n = 1 - 3*Sum_{n>=1} x^(n^2).
(5) Sum_{n=-oo..+oo} x^(n^2) / (1 + 5*x^n*A(x))^(n+1) = 1 - 3*Sum_{n>=1} x^(n^2).
(6) Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 - 5*x^n*A(x))^n = 0.

A370036 Expansion of g.f. A(x) satisfying Sum_{n=-oo..+oo} (x^n - 6*A(x))^n = 1 - 4*Sum_{n>=1} x^(n^2).

Original entry on oeis.org

1, 6, 35, 199, 1117, 6335, 37222, 230809, 1515784, 10423684, 73758799, 529151547, 3815582934, 27567473744, 199625904531, 1451286365478, 10610026385893, 78068267016226, 578088243024187, 4304808678569939, 32204405165738517, 241805832191132439, 1820963567348143772
Offset: 1

Views

Author

Paul D. Hanna, Feb 10 2024

Keywords

Comments

A related function is theta_3(x) = 1 + 2*Sum_{n>=1} x^(n^2).

Examples

			G.f.: A(x) = x + 6*x^2 + 35*x^3 + 199*x^4 + 1117*x^5 + 6335*x^6 + 37222*x^7 + 230809*x^8 + 1515784*x^9 + 10423684*x^10 + 73758799*x^11 + 529151547*x^12 + ...
where
Sum_{n=-oo..+oo} (x^n - 6*A(x))^n = 1 - 4*x - 4*x^4 - 4*x^9 - 4*x^16 - 4*x^25 - 4*x^36 - 4*x^49 - ...
SPECIAL VALUES.
(V.1) Let A = A(exp(-Pi)) = 0.05816104948088020874729529058423242784366544822359858088...
then Sum_{n=-oo..+oo} (exp(-n*Pi) - 6*A)^n = 3 - 2*Pi^(1/4)/gamma(3/4) = 0.82713037757338397...
(V.2) Let A = A(exp(-2*Pi)) = 0.001888597166059649200752082246148944967408910981759517793...
then Sum_{n=-oo..+oo} (exp(-2*n*Pi) - 6*A)^n = 3 - 2*sqrt(2 + sqrt(2))/2 * Pi^(1/4)/gamma(3/4) = 0.9925302290245218...
(V.3) Let A = A(-exp(-Pi)) = -0.03427512499419794844050440831018295417511284891315471397...
then Sum_{n=-oo..+oo} ((-1)^n*exp(-n*Pi) - 6*A)^n = 3 - 2*(Pi/2)^(1/4)/gamma(3/4) = 1.172841723687766...
(V.4) Let A = A(-exp(-2*Pi)) = -0.001846744216948148769402996728724142172026226548695349349...
then Sum_{n=-oo..+oo} ((-1)^n*exp(-2*n*Pi) - 6*A)^n = 3 - 2*2^(1/8)*(Pi/2)^(1/4)/gamma(3/4) = 1.007469770878185...

Links

Crossrefs

Cf. A370041, A370030, A370031, A355868, A370033, A370034, A370035, A370037, A370038, A370039, A370043.

Programs

  • PARI
    {a(n) = my(A=[0,1]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff( sum(m=-#A,#A, (x^m - 6*Ser(A))^m ) - 1 + 4*sum(m=1,#A, x^(m^2) ), #A-1)/6 ); 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} (x^n - 6*A(x))^n = 1 - 4*Sum_{n>=1} x^(n^2).
(2) Sum_{n=-oo..+oo} x^n * (x^n + 6*A(x))^(n-1) = 1 - 4*Sum_{n>=1} x^(n^2).
(3) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n - 6*A(x))^n = 0.
(4) Sum_{n=-oo..+oo} x^(n^2) / (1 - 6*x^n*A(x))^n = 1 - 4*Sum_{n>=1} x^(n^2).
(5) Sum_{n=-oo..+oo} x^(n^2) / (1 + 6*x^n*A(x))^(n+1) = 1 - 4*Sum_{n>=1} x^(n^2).
(6) Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 - 6*x^n*A(x))^n = 0.

A370037 Expansion of g.f. A(x) satisfying Sum_{n=-oo..+oo} (x^n - 7*A(x))^n = 1 - 5*Sum_{n>=1} x^(n^2).

Original entry on oeis.org

1, 7, 48, 323, 2150, 14349, 97431, 681857, 4956116, 37422943, 291763607, 2327820547, 18848767552, 153975563934, 1264733865463, 10431060837749, 86368123241833, 718121985169658, 5997857713743011, 50325664101701349, 424138198629299217, 3589151537280637957, 30481898682409007792
Offset: 1

Views

Author

Paul D. Hanna, Feb 10 2024

Keywords

Comments

A related function is theta_3(x) = 1 + 2*Sum_{n>=1} x^(n^2).

Examples

			G.f.: A(x) = x + 7*x^2 + 48*x^3 + 323*x^4 + 2150*x^5 + 14349*x^6 + 97431*x^7 + 681857*x^8 + 4956116*x^9 + 37422943*x^10 + 291763607*x^11 + ...
where
Sum_{n=-oo..+oo} (x^n - 7*A(x))^n = 1 - 5*x - 5*x^4 - 5*x^9 - 5*x^16 - 5*x^25 - 5*x^36 - 5*x^49 - ...
SPECIAL VALUES.
(V.1) Let A = A(exp(-Pi)) = 0.06174306640715063509845961016774795661670689719654375131...
then Sum_{n=-oo..+oo} (exp(-n*Pi) - 7*A)^n = (7 - 5*Pi^(1/4)/gamma(3/4))/2 = 0.78391297196672996...
(V.2) Let A = A(exp(-2*Pi)) = 0.001892170701611855386420113452656768397809538392272023405...
then Sum_{n=-oo..+oo} (exp(-2*n*Pi) - 7*A)^n = (7 - 5*sqrt(2 + sqrt(2))/2 * Pi^(1/4)/gamma(3/4))/2 = 0.99066278628065227...
(V.3) Let A = A(-exp(-Pi)) = -0.03314064170176376172583062314299135400117746157373562359...
then Sum_{n=-oo..+oo} ((-1)^n*exp(-n*Pi) - 7*A)^n = (7 - 5*(Pi/2)^(1/4)/gamma(3/4))/2 = 1.2160521546097079...
(V.4) Let A = A(-exp(-2*Pi)) = -0.001843340051041967867985717685295689652563446679869985649...
then Sum_{n=-oo..+oo} ((-1)^n*exp(-2*n*Pi) - 7*A)^n = (7 - 5*2^(1/8)*(Pi/2)^(1/4)/gamma(3/4))/2 = 1.00933721359773216...

Links

Crossrefs

Cf. A370041, A370030, A370031, A355868, A370033, A370034, A370035, A370036, A370038, A370039, A370043.

Programs

  • PARI
    {a(n) = my(A=[0,1]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff( sum(m=-#A,#A, (x^m - 7*Ser(A))^m ) - 1 + 5*sum(m=1,#A, x^(m^2) ), #A-1)/7 ); 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} (x^n - 7*A(x))^n = 1 - 5*Sum_{n>=1} x^(n^2).
(2) Sum_{n=-oo..+oo} x^n * (x^n + 7*A(x))^(n-1) = 1 - 5*Sum_{n>=1} x^(n^2).
(3) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n - 7*A(x))^n = 0.
(4) Sum_{n=-oo..+oo} x^(n^2) / (1 - 7*x^n*A(x))^n = 1 - 5*Sum_{n>=1} x^(n^2).
(5) Sum_{n=-oo..+oo} x^(n^2) / (1 + 7*x^n*A(x))^(n+1) = 1 - 5*Sum_{n>=1} x^(n^2).
(6) Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 - 7*x^n*A(x))^n = 0.

A370038 Expansion of g.f. A(x) satisfying Sum_{n=-oo..+oo} (x^n - 8*A(x))^n = 1 - 6*Sum_{n>=1} x^(n^2).

Original entry on oeis.org

1, 8, 63, 489, 3761, 28911, 224174, 1768801, 14298852, 118834966, 1014912939, 8876489811, 79106007766, 714758437500, 6521121292423, 59905861779190, 553172777516749, 5129986605394544, 47761053650028335, 446350549038171483, 4186889953961917077, 39416115485839527945
Offset: 1

Views

Author

Paul D. Hanna, Feb 10 2024

Keywords

Comments

A related function is theta_3(x) = 1 + 2*Sum_{n>=1} x^(n^2).

Examples

			G.f.: A(x) = x + 8*x^2 + 63*x^3 + 489*x^4 + 3761*x^5 + 28911*x^6 + 224174*x^7 + 1768801*x^8 + 14298852*x^9 + 118834966*x^10 + 1014912939*x^11 + ...
where
Sum_{n=-oo..+oo} (x^n - 8*A(x))^n = 1 - 6*x - 6*x^4 - 6*x^9 - 6*x^16 - 6*x^25 - 6*x^36 - 6*x^49 - ...
SPECIAL VALUES.
(V.1) Let A = A(exp(-Pi)) = 0.06579433445460281447496748523290398966344297589844019028...
then Sum_{n=-oo..+oo} (exp(-n*Pi) - 8*A)^n = 4 - 3*Pi^(1/4)/gamma(3/4) = 0.740695566360075956...
(V.2) Let A = A(exp(-2*Pi)) = 0.001895757786183755555448115532175643265455444051246465664...
then Sum_{n=-oo..+oo} (exp(-2*n*Pi) - 8*A)^n = 4 - 3*sqrt(2 + sqrt(2))/2 * Pi^(1/4)/gamma(3/4) = 0.98879534353678272...
(V.3) Let A = A(-exp(-Pi)) = -0.03207876150064786089070312769117792591667175850120792604...
then Sum_{n=-oo..+oo} ((-1)^n*exp(-n*Pi) - 8*A)^n = 4 - 3*(Pi/2)^(1/4)/gamma(3/4) = 1.25926258553164953...
(V.4) Let A = A(-exp(-2*Pi)) = -0.001839948412029108042031275075360099309960919616491079407...
then Sum_{n=-oo..+oo} ((-1)^n*exp(-2*n*Pi) - 8*A)^n = 4 - 3*2^(1/8)*(Pi/2)^(1/4)/gamma(3/4) = 1.01120465631727859...

Links

Crossrefs

Cf. A370041, A370030, A370031, A355868, A370033, A370034, A370035, A370036, A370037, A370039, A370043.

Programs

  • PARI
    {a(n) = my(A=[0,1]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff( sum(m=-#A,#A, (x^m - 8*Ser(A))^m ) - 1 + 6*sum(m=1,#A, x^(m^2) ), #A-1)/8 ); 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} (x^n - 8*A(x))^n = 1 - 6*Sum_{n>=1} x^(n^2).
(2) Sum_{n=-oo..+oo} x^n * (x^n + 8*A(x))^(n-1) = 1 - 6*Sum_{n>=1} x^(n^2).
(3) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n - 8*A(x))^n = 0.
(4) Sum_{n=-oo..+oo} x^(n^2) / (1 - 8*x^n*A(x))^n = 1 - 6*Sum_{n>=1} x^(n^2).
(5) Sum_{n=-oo..+oo} x^(n^2) / (1 + 8*x^n*A(x))^(n+1) = 1 - 6*Sum_{n>=1} x^(n^2).
(6) Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 - 8*x^n*A(x))^n = 0.
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