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.

Showing 1-3 of 3 results.

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.

A370153 Expansion of g.f. (theta_3(x) - 1)/2 * Product_{n>=1} (1 - x^(4*n-2)) / (1 - x^(4*n)).

Original entry on oeis.org

1, 0, -1, 1, 1, -1, -2, 1, 4, -2, -5, 3, 6, -4, -9, 6, 13, -8, -17, 11, 21, -15, -28, 19, 39, -25, -49, 33, 60, -42, -78, 53, 101, -68, -125, 87, 153, -108, -192, 134, 241, -167, -295, 207, 357, -255, -438, 311, 540, -380, -652, 465, 781, -563, -946, 678, 1145, -819, -1368, 986, 1627
Offset: 1

Views

Author

Paul D. Hanna, Feb 10 2024

Keywords

Comments

Column 0 of triangle A370041. The g.f. of triangle A370041, G(x,y), satisfies Sum_{n=-oo..+oo} (x^n - y*G(x,y))^n = 1 - (y-2)*Sum_{n>=1} x^(n^2). The g.f. of this sequence is G(x,y) at y = 0.

Examples

			G.f.: A(x) = x - x^3 + x^4 + x^5 - x^6 - 2*x^7 + x^8 + 4*x^9 - 2*x^10 - 5*x^11 + 3*x^12 + 6*x^13 - 4*x^14 - 9*x^15 + 6*x^16 + 13*x^17 - 8*x^18 + ...
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 + x^2 + x^6 + x^12 + x^20 + x^30 + x^42 + ...

Links

Crossrefs

Cf. A370041, A370154, A370155.
Cf. A370150 (dual).

Programs

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

Formula

a(n) = A370041(n,0) for n >= 1.
G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) A(x) = (theta_3(x) - 1)/2 * Product_{n>=1} (1 - x^(4*n-2))/(1 - x^(4*n)).
(2) A(x) = P(x)/Q(x) where P(x) = Sum_{n>=1} x^(n^2) and Q(x) = Sum_{n>=0} x^(n*(n+1)).
(3) A(x) = G(x,0) where G(x,y) is the g.f. of triangle A370041 (see comment).

A370155 Column 2 of triangle A370041.

Original entry on oeis.org

1, 0, -6, 19, 17, -98, 39, 291, -264, -627, 773, 1525, -2000, -3895, 5606, 8784, -14325, -18125, 32856, 37727, -72557, -77316, 157044, 150945, -326602, -286320, 655760, 536800, -1287225, -986584, 2474251, 1770786, -4650780, -3122412, 8570426, 5430403, -15530336, -9307817, 27707720
Offset: 3

Views

Author

Paul D. Hanna, Feb 10 2024

Keywords

Comments

The g.f. of triangle A370041, G(x,y), satisfies Sum_{n=-oo..+oo} (x^n - y*G(x,y))^n = 1 - (y-2)*Sum_{n>=1} x^(n^2).

Links

Crossrefs

Cf. A370041, A370153, A370154.

Formula

a(n) = A370041(n,2) for n >= 3.
Showing 1-3 of 3 results.

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