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 1-2 of 2 results.

A356777 G.f.: Sum_{n=-oo..+oo} x^(n^2) * C(x)^(2*n-1), where C(x) = 1 + x*C(x)^2 is a g.f. of the Catalan numbers (A000108).

Original entry on oeis.org

1, 1, -3, 0, 1, -5, 5, 0, 0, 1, -7, 14, -7, 0, 0, 0, 1, -9, 27, -30, 9, 0, 0, 0, 0, 1, -11, 44, -77, 55, -11, 0, 0, 0, 0, 0, 1, -13, 65, -156, 182, -91, 13, 0, 0, 0, 0, 0, 0, 1, -15, 90, -275, 450, -378, 140, -15, 0, 0, 0, 0, 0, 0, 0, 1, -17, 119, -442, 935, -1122, 714, -204, 17, 0, 0, 0, 0, 0, 0, 0, 0, 1, -19, 152, -665, 1729, -2717, 2508, -1254, 285, -19
Offset: 0

Views

Author

Paul D. Hanna, Sep 08 2022

Keywords

Examples

			G.f.: A(x) = 1 + x - 3*x^2 + x^4 - 5*x^5 + 5*x^6 + x^9 - 7*x^10 + 14*x^11 - 7*x^12 + x^16 - 9*x^17 + 27*x^18 - 30*x^19 + 9*x^20 + x^25 - 11*x^26 + 44*x^27 - 77*x^28 + 55*x^29 - 11*x^30 + x^36 - 13*x^37 + 65*x^38 - 156*x^39 + 182*x^40 - 91*x^41 + 13*x^42 + x^49 - 15*x^50 + 90*x^51 - 275*x^52 + 450*x^53 - 378*x^54 + 140*x^55 - 15*x^56 + ...
such that
A(x) = ... + x^16/C(x)^9 + x^9/C(x)^7 + x^4/C(x)^5 + x/C(x)^3 + 1/C(x) + x*C(x) + x^4*C(x)^3 + x^9*C(x)^5 + x^16*C(x)^7 + x^25*C(x)^9 + ... + x^(n^2)*C^(2*n-1) + ...
where the Catalan function C(x) = (1 - sqrt(1-4*x))/(2*x) begins
C(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 + 429*x^7 + 1430*x^8 + 4862*x^9 + ... + A000108(n)*x^n + ...
RELATED TABLE.
This sequence may be written in the form of an irregular triangle:
1,
1, -3, 0,
1, -5, 5, 0, 0,
1, -7, 14, -7, 0, 0, 0,
1, -9, 27, -30, 9, 0, 0, 0, 0,
1, -11, 44, -77, 55, -11, 0, 0, 0, 0, 0,
1, -13, 65, -156, 182, -91, 13, 0, 0, 0, 0, 0, 0,
1, -15, 90, -275, 450, -378, 140, -15, 0, 0, 0, 0, 0, 0, 0,
1, -17, 119, -442, 935, -1122, 714, -204, 17, 0, 0, 0, 0, 0, 0, 0, 0,
1, -19, 152, -665, 1729, -2717, 2508, -1254, 285, -19, 0, 0, 0, 0, 0, 0, 0, 0, 0,
...
Compare the above construction to triangle A082985.

Links

Crossrefs

Cf. A355341, A355342, A355345, A356778, A082985, A000108.

Programs

  • PARI
    /* By Definition: */
{a(n) = my(A, C = 1/x*serreverse(x-x^2 +O(x^(n+2))), M=ceil(sqrt(n+1)));
A = sum(m=-M, M, x^(m^2) * C^(2*m-1) ); polcoeff(A, n)}
for(n=0, 90, print1(a(n), ", "))
  • PARI
    /* Without Using Catalan Series */
{a(n) = my(A, M=ceil(sqrt(n+1)));
A = sum(m=0, M, sum(k=0, 2*m, (-1)^k*binomial(2*m-k, k)*(2*m+1)/(2*m-2*k+1) * x^(m^2 + k) ) +x*O(x^n)); polcoeff(A, n)}
for(n=0, 90, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n may be obtained from the following expressions; here, C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108).
(1) A(x) = Sum_{n=-oo..+oo} x^(n^2) * C(x)^(2*n-1).
(2) A(x) = 1/C(x) * Product_{n>=1} (1 + x^(2*n-1)*C(x)^2) * (1 + x^(2*n-1)/C(x)^2) * (1 - x^(2*n)), by the Jacobi triple product identity.
(3) A(x) = 1/C(x) + Sum_{n>=1} x^(n^2) * (C(x)^(2*n-1) + 1/C(x)^(2*n+1)).
(4) A(x) = Sum_{n>=0} Sum_{k=0..n} (-1)^k * binomial(2*n-k, k) * (2*n+1)/(2*n-2*k+1) * x^(n^2 + k).

A356779 G.f.: Sum_{n=-oo..+oo} x^(n^2) * C(x)^(6*n-9), where C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108).

Original entry on oeis.org

1, -7, 9, 60, -265, 429, -189, -812, 2925, -5732, 6980, -4824, -198, 10010, -32298, 69768, -104651, 107373, -72435, 26422, 19656, -115011, 361763, -834900, 1427679, -1797817, 1641447, -1057446, 454155, -69564, -298980, 1307448, -4102104, 9924525, -18599295
Offset: 0

Views

Author

Paul D. Hanna, Sep 08 2022

Keywords

Examples

			G.f.: A(x) = 1 - 7*x + 9*x^2 + 60*x^3 - 265*x^4 + 429*x^5 - 189*x^6 - 812*x^7 + 2925*x^8 - 5732*x^9 + 6980*x^10 - 4824*x^11 - 198*x^12 + 10010*x^13 - 32298*x^14 + 69768*x^15 - 104651*x^16 + 107373*x^17 - 72435*x^18 + 26422*x^19 + 19656*x^20 - 115011*x^21 + 361763*x^22 - 834900*x^23 + 1427679*x^24 - 1797817*x^25 + ...
such that
A(x) = ... + x^16/C(x)^33 + x^9/C(x)^27 + x^4/C(x)^21 + x/C(x)^15 + 1/C(x)^9 + x/C(x)^3 + x^4*C(x)^3 + x^9*C(x)^9 + x^16*C(x)^15 + x^25*C(x)^21 + ... + x^(n^2)*C(x)^(6*n-9) + ...
where the Catalan function C(x) = (1 - sqrt(1-4*x))/(2*x) begins
C(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 + 429*x^7 + 1430*x^8 + 4862*x^9 + ... + A000108(n)*x^n + ...
RELATED TABLE.
This sequence may be written in the form of an irregular triangle that begins:
1,
-7, 9, 60,
-265, 429, -189, -812, 2925,
-5732, 6980, -4824, -198, 10010, -32298, 69768,
-104651, 107373, -72435, 26422, 19656, -115011, 361763, -834900, 1427679,
-1797817, 1641447, -1057446, 454155, -69564, -298980, 1307448, -4102104, 9924525, -18599295, 26936910,
-29910464, 25109975, -15599955, 6941244, -2013544, -324558, 3717882, -14942570, 46955661, -117679100, 236030652, -378658800, 483841800,
...

Links

Crossrefs

Cf. A356777, A356778, A355341, A355345, A034807, A000108.

Programs

  • PARI
    /* By Definition: */
{a(n) = my(A, C=1/x*serreverse(x-x^2 +O(x^(n+2))), M=ceil(sqrt(n+1)));
A = sum(m=-M, M, x^(m^2) * C^(6*m-9) ); polcoeff(A, n)}
for(n=0, 50, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n may be obtained from the following expressions; here, C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108).
(1) A(x) = Sum_{n=-oo..+oo} x^(n^2) * C(x)^(6*n-9).
(2) A(x) = 1/C(x)^9 * Product_{n>=1} (1 + x^(2*n-1)*C(x)^6) * (1 + x^(2*n-1)/C(x)^6) * (1 - x^(2*n)), by the Jacobi triple product identity.
(3) A(x) = 1/C(x)^9 + Sum_{n>=1} x^(n^2) * (C(x)^(6*n-9) + 1/C(x)^(6*n+9)).
Showing 1-2 of 2 results.

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