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 1-4 of 4 results.

A370141 Expansion of g.f. A(x) satisfying Sum_{n>=0} Product_{k=1..n} (x^k + A(x)) = 1 + 2*Sum_{n>=1} x^(n*(n+1)/2).

Original entry on oeis.org

1, -2, 4, -8, 14, -22, 28, -14, -80, 420, -1430, 4128, -10798, 26176, -59114, 123442, -232240, 365888, -355616, -475892, 4318112, -17471288, 56635490, -163101656, 432173038, -1067080032, 2456709054, -5216642696, 9906435640, -15415122000, 12937725806, 33034018944, -238942986520
Offset: 1

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Author

Paul D. Hanna, Feb 14 2024

Keywords

Examples

			G.f.: A(x) = x - 2*x^2 + 4*x^3 - 8*x^4 + 14*x^5 - 22*x^6 + 28*x^7 - 14*x^8 - 80*x^9 + 420*x^10 - 1430*x^11 + 4128*x^12 + ...
Let Q(x) = 1 + 2*Sum_{n>=1} x^(n*(n+1)/2)
then A = A(x) satisfies
(1) Q(x) = 1 + (x + A) + (x + A)*(x^2 + A) + (x + A)*(x^2 + A)*(x^3 + A) + (x + A)*(x^2 + A)*(x^3 + A)*(x^4 + A) + (x + A)*(x^2 + A)*(x^3 + A)*(x^4 + A)*(x^5 + A) + ...
also
(2) Q(x) = 1/(1 - A) + x/((1 - A)*(1 - x*y*A)) + x^3/((1 - A)*(1 - x*y*A)*(1 - x^2*y*A)) + x^6/((1 - A)*(1 - x*y*A)*(1 - x^2*y*A)*(1 - x^3*y*A)) + x^10/((1 - A)*(1 - x*y*A)*(1 - x^2*y*A)*(1 - x^3*y*A)*(1 - x^4*y*A)) + ...
Further, A = A(x) satisfies the continued fraction given by
(3) Q(x) = 1/(1 - (x + A)/(1 + x + A - (x^2 + A)/(1 + x^2 + A - (x^3 + A)/(1 + x^3 + A - (x^4 + A)/(1 + x^4 + A - (x^5 + A)/(1 + x^5 + A - (x^6 + A)/(1 + x^6 + A - (x^7 + A)/(1 - ...)))))))).
where
Q(x) = 1 + 2*x + 2*x^3 + 2*x^6 + 2*x^10 + 2*x^15 + 2*x^21 + ... + 2*x^(n*(n+1)/2) + ...

Links

Crossrefs

Cf. A370140, A370142, A370143, A370144.

Programs

  • PARI
    {a(n,y=1) = my(A=[0,1]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff( (sum(m=1,#A, prod(k=1,m, x^k + y*Ser(A) ) ) - (y+1)*sum(m=1,sqrtint(2*#A+1), x^(m*(m+1)/2) ) )/(-y), #A-1)  ); H=A; 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.
Let Q(x) = 1 + 2*Sum_{n>=1} x^(n*(n+1)/2), then
(1) Q(x) = Sum_{n>=0} Product_{k=1..n} (x^k + A(x)).
(2) Q(x) = Sum_{n>=0} x^(n*(n+1)/2) / Product_{k=0..n} (1 - x^k * A(x)).
(3) Q(x) = 1/(1 - F(1)), where F(n) = (x^n + A(x))/(1 + x^n + A(x) - F(n+1)), a continued fraction.

A370142 Expansion of g.f. A(x) satisfying Sum_{n>=0} Product_{k=1..n} (x^k + 2*A(x)) = 1 + 3*Sum_{n>=1} x^(n*(n+1)/2).

Original entry on oeis.org

1, -3, 9, -27, 78, -219, 591, -1500, 3420, -6153, 3315, 44466, -324276, 1627002, -7069893, 28345875, -107618916, 391749108, -1375531170, 4669215090, -15311251593, 48316101369, -145501913850, 411323278248, -1053727809204, 2226156968586, -2433380638410, -10543933246791
Offset: 1

Views

Author

Paul D. Hanna, Feb 14 2024

Keywords

Examples

			G.f.: A(x) = x - 3*x^2 + 9*x^3 - 27*x^4 + 78*x^5 - 219*x^6 + 591*x^7 - 1500*x^8 + 3420*x^9 - 6153*x^10 + 3315*x^11 + 44466*x^12 + ...
Let Q(x) = 1 + 3*Sum_{n>=1} x^(n*(n+1)/2)
then A = A(x) satisfies
(1) Q(x) = 1 + (x + 2*A) + (x + 2*A)*(x^2 + 2*A) + (x + 2*A)*(x^2 + 2*A)*(x^3 + 2*A) + (x + 2*A)*(x^2 + 2*A)*(x^3 + 2*A)*(x^4 + 2*A) + (x + 2*A)*(x^2 + 2*A)*(x^3 + 2*A)*(x^4 + 2*A)*(x^5 + 2*A) + ...
also
(2) Q(x) = 1/(1 - 2*A) + x/((1 - 2*A)*(1 - x*2*A)) + x^3/((1 - 2*A)*(1 - x*2*A)*(1 - x^2*2*A)) + x^6/((1 - 2*A)*(1 - x*2*A)*(1 - x^2*2*A)*(1 - x^3*2*A)) + x^10/((1 - 2*A)*(1 - x*2*A)*(1 - x^2*2*A)*(1 - x^3*2*A)*(1 - x^4*2*A)) + ...
Further, A = A(x) satisfies the continued fraction given by
(3) Q(x) = 1/(1 - (x + 2*A)/(1 + x + 2*A - (x^2 + 2*A)/(1 + x^2 + 2*A - (x^3 + 2*A)/(1 + x^3 + 2*A - (x^4 + 2*A)/(1 + x^4 + 2*A - (x^5 + 2*A)/(1 + x^5 + 2*A - (x^6 + 2*A)/(1 + x^6 + 2*A - (x^7 + 2*A)/(1 - ...)))))))).
where
Q(x) = 1 + 3*x + 3*x^3 + 3*x^6 + 3*x^10 + 3*x^15 + 3*x^21 + ... + 3*x^(n*(n+1)/2) + ...

Links

Crossrefs

Cf. A370140, A370141, A370143, A370144.

Programs

  • PARI
    {a(n,y=2) = my(A=[0,1]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff( (sum(m=1,#A, prod(k=1,m, x^k + y*Ser(A) ) ) - (y+1)*sum(m=1,sqrtint(2*#A+1), x^(m*(m+1)/2) ) )/(-y), #A-1)  ); H=A; 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.
Let Q(x) = 1 + 3*Sum_{n>=1} x^(n*(n+1)/2), then
(1) Q(x) = Sum_{n>=0} Product_{k=1..n} (x^k + 2*A(x)).
(2) Q(x) = Sum_{n>=0} x^(n*(n+1)/2) / Product_{k=0..n} (1 - x^k * 2*A(x)).
(3) Q(x) = 1/(1 - F(1)), where F(n) = (x^n + 2*A(x))/(1 + x^n + 2*A(x) - F(n+1)), a continued fraction.

A370143 Expansion of g.f. A(x) satisfying Sum_{n>=0} Product_{k=1..n} (x^k + 3*A(x)) = 1 + 4*Sum_{n>=1} x^(n*(n+1)/2).

Original entry on oeis.org

1, -4, 16, -64, 252, -980, 3752, -14076, 51384, -180488, 597812, -1788936, 4284828, -3665376, -47694524, 475585724, -3186717720, 18465627936, -98993741736, 504886869432, -2484393916472, 11887606679816, -55566604265244, 254411449305096, -1142492183274444, 5033985939170544
Offset: 1

Views

Author

Paul D. Hanna, Feb 14 2024

Keywords

Examples

			G.f.: A(x) = x - 4*x^2 + 16*x^3 - 64*x^4 + 252*x^5 - 980*x^6 + 3752*x^7 - 14076*x^8 + 51384*x^9 - 180488*x^10 + 597812*x^11 - 1788936*x^12 + ...
Let Q(x) = 1 + 4*Sum_{n>=1} x^(n*(n+1)/2)
then A = A(x) satisfies
(1) Q(x) = 1 + (x + 3*A) + (x + 3*A)*(x^2 + 3*A) + (x + 3*A)*(x^2 + 3*A)*(x^3 + 3*A) + (x + 3*A)*(x^2 + 3*A)*(x^3 + 3*A)*(x^4 + 3*A) + (x + 3*A)*(x^2 + 3*A)*(x^3 + 3*A)*(x^4 + 3*A)*(x^5 + 3*A) + ...
also
(2) Q(x) = 1/(1 - 3*A) + x/((1 - 3*A)*(1 - x*3*A)) + x^3/((1 - 3*A)*(1 - x*3*A)*(1 - x^2*3*A)) + x^6/((1 - 3*A)*(1 - x*3*A)*(1 - x^2*3*A)*(1 - x^3*3*A)) + x^10/((1 - 3*A)*(1 - x*3*A)*(1 - x^2*3*A)*(1 - x^3*3*A)*(1 - x^4*3*A)) + ...
Further, A = A(x) satisfies the continued fraction given by
(3) Q(x) = 1/(1 - (x + 3*A)/(1 + x + 3*A - (x^2 + 3*A)/(1 + x^2 + 3*A - (x^3 + 3*A)/(1 + x^3 + 3*A - (x^4 + 3*A)/(1 + x^4 + 3*A - (x^5 + 3*A)/(1 + x^5 + 3*A - (x^6 + 3*A)/(1 + x^6 + 3*A - (x^7 + 3*A)/(1 - ...)))))))).
where
Q(x) = 1 + 4*x + 4*x^3 + 4*x^6 + 4*x^10 + 4*x^15 + 4*x^21 + ... + 4*x^(n*(n+1)/2) + ...

Links

Crossrefs

Cf. A370140, A370141, A370142, A370144.

Programs

  • PARI
    {a(n,y=3) = my(A=[0,1]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff( (sum(m=1,#A, prod(k=1,m, x^k + y*Ser(A) ) ) - (y+1)*sum(m=1,sqrtint(2*#A+1), x^(m*(m+1)/2) ) )/(-y), #A-1)  ); H=A; 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.
Let Q(x) = 1 + 4*Sum_{n>=1} x^(n*(n+1)/2), then
(1) Q(x) = Sum_{n>=0} Product_{k=1..n} (x^k + 3*A(x)).
(2) Q(x) = Sum_{n>=0} x^(n*(n+1)/2) / Product_{k=0..n} (1 - x^k * 3*A(x)).
(3) Q(x) = 1/(1 - F(1)), where F(n) = (x^n + 3*A(x))/(1 + x^n + 3*A(x) - F(n+1)), a continued fraction.

A370144 Expansion of g.f. A(x) satisfying Sum_{n>=0} Product_{k=1..n} (x^k + 4*A(x)) = 1 + 5*Sum_{n>=1} x^(n*(n+1)/2).

Original entry on oeis.org

1, -5, 25, -125, 620, -3055, 14935, -72320, 346120, -1632435, 7555615, -34103940, 148616480, -614229500, 2321723005, -7293541865, 11847720800, 80050256480, -1182000043580, 10263664188460, -75379977023875, 508865668922995, -3262422374486260, 20187692749822600, -121673379897635840
Offset: 1

Views

Author

Paul D. Hanna, Feb 14 2024

Keywords

Examples

			G.f.: A(x) = x - 5*x^2 + 25*x^3 - 125*x^4 + 620*x^5 - 3055*x^6 + 14935*x^7 - 72320*x^8 + 346120*x^9 - 1632435*x^10 + 7555615*x^11 + ...
Let Q(x) = 1 + 5*Sum_{n>=1} x^(n*(n+1)/2)
then A = A(x) satisfies
(1) Q(x) = 1 + (x + 4*A) + (x + 4*A)*(x^2 + 4*A) + (x + 4*A)*(x^2 + 4*A)*(x^3 + 4*A) + (x + 4*A)*(x^2 + 4*A)*(x^3 + 4*A)*(x^4 + 4*A) + (x + 4*A)*(x^2 + 4*A)*(x^3 + 4*A)*(x^4 + 4*A)*(x^5 + 4*A) + ...
also
(2) Q(x) = 1/(1 - 4*A) + x/((1 - 4*A)*(1 - x*4*A)) + x^3/((1 - 4*A)*(1 - x*4*A)*(1 - x^2*4*A)) + x^6/((1 - 4*A)*(1 - x*4*A)*(1 - x^2*4*A)*(1 - x^3*4*A)) + x^10/((1 - 4*A)*(1 - x*4*A)*(1 - x^2*4*A)*(1 - x^3*4*A)*(1 - x^4*4*A)) + ...
Further, A = A(x) satisfies the continued fraction given by
(3) Q(x) = 1/(1 - (x + 4*A)/(1 + x + 4*A - (x^2 + 4*A)/(1 + x^2 + 4*A - (x^3 + 4*A)/(1 + x^3 + 4*A - (x^4 + 4*A)/(1 + x^4 + 4*A - (x^5 + 4*A)/(1 + x^5 + 4*A - (x^6 + 4*A)/(1 + x^6 + 4*A - (x^7 + 4*A)/(1 - ...)))))))).
where
Q(x) = 1 + 5*x + 5*x^3 + 5*x^6 + 5*x^10 + 5*x^15 + 5*x^21 + ... + 5*x^(n*(n+1)/2) + ...

Links

Crossrefs

Cf. A370140, A370141, A370142, A370143.

Programs

  • PARI
    {a(n,y=4) = my(A=[0,1]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff( (sum(m=1,#A, prod(k=1,m, x^k + y*Ser(A) ) ) - (y+1)*sum(m=1,sqrtint(2*#A+1), x^(m*(m+1)/2) ) )/(-y), #A-1)  ); H=A; 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.
Let Q(x) = 1 + 5*Sum_{n>=1} x^(n*(n+1)/2), then
(1) Q(x) = Sum_{n>=0} Product_{k=1..n} (x^k + 4*A(x)).
(2) Q(x) = Sum_{n>=0} x^(n*(n+1)/2) / Product_{k=0..n} (1 - x^k * 4*A(x)).
(3) Q(x) = 1/(1 - F(1)), where F(n) = (x^n + 4*A(x))/(1 + x^n + 4*A(x) - F(n+1)), a continued fraction.
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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