A370040 -id:A370040 - cp's OEIS frontend

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

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

Paul D. Hanna, Feb 10 2024

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.

			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;
  ...

Paul D. Hanna, Table of n, a(n) for n = 1..2556

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).

/* 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(""))

/* 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(""))

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.

A370020 Table in which the g.f. of row n, R(n,x), satisfies Sum_{k=-oo..+oo} (-1)^k * (x^k + nR(n,x))^k = 1 + (n+2)Sum_{k>=1} (-1)^k * x^(k^2), for n >= 1, as read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 4, 1, 3, 7, 9, 1, 4, 12, 25, 22, 1, 5, 19, 53, 85, 63, 1, 6, 28, 99, 234, 301, 155, 1, 7, 39, 169, 529, 1041, 1086, 415, 1, 8, 52, 269, 1054, 2853, 4711, 3927, 1124, 1, 9, 67, 405, 1917, 6667, 15566, 21573, 14328, 2957, 1, 10, 84, 583, 3250, 13893, 42627, 85879, 99484, 52724, 8047, 1, 11, 103, 809, 5209, 26541, 101830, 275211, 477716, 461657, 194915, 21817

Offset: 1

Paul D. Hanna, Feb 09 2024

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

			This table of coefficients T(n,k) of x^k in R(n,x), n >= 1, k >= 1, begins:
A370021: [1,  1,   4,    9,    22,     63,     155,      415, ...];
A370022: [1,  2,   7,   25,    85,    301,    1086,     3927, ...];
A370023: [1,  3,  12,   53,   234,   1041,    4711,    21573, ...];
A370024: [1,  4,  19,   99,   529,   2853,   15566,    85879, ...];
A370025: [1,  5,  28,  169,  1054,   6667,   42627,   275211, ...];
A370026: [1,  6,  39,  269,  1917,  13893,  101830,   753255, ...];
A370027: [1,  7,  52,  405,  3250,  26541,  219311,  1828657, ...];
A370028: [1,  8,  67,  583,  5209,  47341,  435366,  4039863, ...];
A370029: [1,  9,  84,  809,  7974,  79863,  809131,  8270199, ...];
A370042: [1, 10, 103, 1089, 11749, 128637, 1423982, 15898231, ...];
...
where the n-th row function R(n,x) satisfies
Sum_{k=-oo..+oo} (-1)^k * (x^k + n*R(n,x))^k = 1 + (n+2)*Sum_{k>=1} (-1)^k * x^(k^2).

Paul D. Hanna, Table of n, a(n) for n = 1..1275
Eric Weisstein's World of Mathematics, Jacobi Theta Functions

Cf. A370021, A370022, A370023, A370024, A370025, A370026, A370027, A370028, A370029, A370042.

Cf. A370040, A370030 (dual table).

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

The n-th row g.f. R(n,x) = Sum_{k>=1} T(n,k)*x^k satisfies the following formulas.
(1) Sum_{k=-oo..+oo} (-1)^k * (x^k + n*R(n,x))^k = 1 + (n+2)*Sum_{k>=1} (-1)^k * x^(k^2).
(2) Sum_{k=-oo..+oo} (-1)^k * x^k * (x^k + n*R(n,x))^(k-1) = 1 + (n+2)*Sum_{k>=1} (-1)^k * 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} (-1)^k * x^(k^2) / (1 + n*R(n,x)*x^k)^k = 1 + (n+2)*Sum_{k>=1} (-1)^k * x^(k^2).
(5) Sum_{k=-oo..+oo} (-1)^k * x^(k^2) / (1 + n*R(n,x)*x^k)^(k+1) = 1 + (n+2)*Sum_{k>=1} (-1)^k * x^(k^2).
(6) Sum_{k=-oo..+oo} (-1)^k * x^(k*(k+1)) / (1 + n*R(n,x)*x^k)^(k+1) = 0.

A370150 Expansion of g.f. (1 - theta_4(x))/2 / Product_{n>=1} (1 - x^(2*n))^3.

Original entry on oeis.org

1, 0, 3, -1, 9, -3, 22, -9, 52, -22, 111, -51, 230, -108, 451, -222, 861, -432, 1587, -819, 2861, -1501, 5028, -2691, 8679, -4707, 14691, -8089, 24492, -13638, 40202, -22653, 65141, -37060, 104199, -59863, 164833, -95484, 257920, -150646, 399681, -235141, 613557, -363543, 933869

Offset: 1

Paul D. Hanna, Feb 10 2024

sign

Column 0 of triangle A370040. The g.f. of triangle A370040, F(x,y), satisfies Sum_{n=-oo..+oo} (-1)^n * (x^n + y*F(x,y))^n = 1 + (y+2)*Sum_{n>=1} (-1)^n * x^(n^2). The g.f. of this sequence is F(x,y) at y = 0.

			G.f.: A(x) = x + 3*x^3 - x^4 + 9*x^5 - 3*x^6 + 22*x^7 - 9*x^8 + 52*x^9 - 22*x^10 + 111*x^11 - 51*x^12 + 230*x^13 - 108*x^14 + 451*x^15 - 222*x^16 + ...
which equals A(x) = P(x) / Q(x)
where
P(x) = x - x^4 + x^9 - x^16 + x^25 - x^36 + x^49 + ...
Q(x) = 1 - 3*x^2 + 5*x^6 - 7*x^12 + 9*x^20 - 11*x^30 + 13*x^42 + ...

Paul D. Hanna, Table of n, a(n) for n = 1..202
Eric Weisstein's World of Mathematics, Jacobi Theta Functions

Cf. A370040, A370151, A370152.

Cf. A370153 (dual).

{a(n) = my(P = sum(m=1,sqrtint(n+1), (-1)^(m-1) * x^(m^2) +x*O(x^n)),
Q = sum(m=0,sqrtint(n+1), (-1)^m * (2*m+1) * x^(m*(m+1)) +x*O(x^n)));
polcoeff(P/Q,n)}
for(n=1,50,print1(a(n),", "))

a(n) = A370040(n,0) for n >= 1.
G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) A(x) = (1 - theta_4(x))/2 / Product_{n>=1} (1 - x^(2*n))^3.
(2) A(x) = P(x)/Q(x) where P(x) = Sum_{n>=1} (-1)^(n-1) * x^(n^2) and Q(x) = Sum_{n>=0} (-1)^n * (2*n+1) * x^(n*(n+1)).
(3) A(x) = F(x,0) where F(x,y) is the g.f. of triangle A370040 (see comment).

cp's OEIS Frontend

A370151 Column 1 of triangle A370040.

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A370152 Column 2 of triangle A370040.

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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.

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A370020 Table in which the g.f. of row n, R(n,x), satisfies Sum_{k=-oo..+oo} (-1)^k * (x^k + n*R(n,x))^k = 1 + (n+2)*Sum_{k>=1} (-1)^k * x^(k^2), for n >= 1, as read by antidiagonals.

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A370150 Expansion of g.f. (1 - theta_4(x))/2 / Product_{n>=1} (1 - x^(2*n))^3.

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

A370020 Table in which the g.f. of row n, R(n,x), satisfies Sum_{k=-oo..+oo} (-1)^k * (x^k + nR(n,x))^k = 1 + (n+2)Sum_{k>=1} (-1)^k * x^(k^2), for n >= 1, as read by antidiagonals.