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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A357223 Coefficients in the power series A(x) such that: x*A(x)^3 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) * A(x)^n.

Original entry on oeis.org

1, 1, 4, 25, 164, 1177, 8887, 69748, 563232, 4649672, 39063521, 332904462, 2870862974, 25005954906, 219675658337, 1944131038267, 17316793719372, 155122164103293, 1396584226654493, 12630315100857638, 114687815080027358, 1045218902425525155, 9557367319452886097
Offset: 0

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Author

Paul D. Hanna, Sep 18 2022

Keywords

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 25*x^3 + 164*x^4 + 1177*x^5 + 8887*x^6 + 69748*x^7 + 563232*x^8 + 4649672*x^9 + 39063521*x^10 + ...
such that
x*A(x)^3 = ... + x^12/A(x)^4 - x^6/A(x)^3 + x^2/A(x)^2 - 1/A(x) + 1 - x^2*A(x) + x^6*A(x)^2 - x^12*A(x)^3 + x^20*A(x)^4 + ... + (-1)^n * x^(n*(n+1)) * A(x)^n + ...

Crossrefs

Cf. A355357, A357221, A357222, A357224, A357225, A357226.

Programs

  • PARI
    {a(n,p=3) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff( x*Ser(A)^p - sum(m=-ceil(sqrt(n)), ceil(sqrt(n)), (-1)^m*x^(m*(m+1))*Ser(A)^m ), #A-1)); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) satisfies:
(1) x*A(x)^3 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) * A(x)^n.
(2) -x*A(x)^4 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) / A(x)^n.
(3) x*A(x)^3 = Product_{n>=1} (1 - x^(2*n)*A(x)) * (1 - x^(2*n-2)/A(x)) * (1 - x^(2*n)), due to the Jacobi triple product identity.
(4) -x*A(x)^4 = Product_{n>=1} (1 - x^(2*n)/A(x)) * (1 - x^(2*n-2)*A(x)) * (1 - x^(2*n)), due to the Jacobi triple product identity.

A357224 Coefficients in the power series A(x) such that: x*A(x)^4 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) * A(x)^n.

Original entry on oeis.org

1, 1, 5, 38, 315, 2855, 27325, 272030, 2788042, 29221793, 311767823, 3374650902, 36968040004, 409076635878, 4565873250981, 51342245169913, 581093383193700, 6614534942714496, 75675364150733073, 869713202188274489, 10036085000519702155, 116238137830534589525
Offset: 0

Views

Author

Paul D. Hanna, Sep 18 2022

Keywords

Examples

			G.f.: A(x) = 1 + x + 5*x^2 + 38*x^3 + 315*x^4 + 2855*x^5 + 27325*x^6 + 272030*x^7 + 2788042*x^8 + 29221793*x^9 + 311767823*x^10 + ...
such that
x*A(x)^4 = ... + x^12/A(x)^4 - x^6/A(x)^3 + x^2/A(x)^2 - 1/A(x) + 1 - x^2*A(x) + x^6*A(x)^2 - x^12*A(x)^3 + x^20*A(x)^4 + ... + (-1)^n * x^(n*(n+1)) * A(x)^n + ...

Crossrefs

Cf. A355357, A357221, A357222, A357223, A357225, A357226.

Programs

  • PARI
    {a(n,p=4) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff( x*Ser(A)^p - sum(m=-ceil(sqrt(n)), ceil(sqrt(n)), (-1)^m*x^(m*(m+1))*Ser(A)^m ), #A-1)); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) satisfies:
(1) x*A(x)^4 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) * A(x)^n.
(2) -x*A(x)^5 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) / A(x)^n.
(3) x*A(x)^4 = Product_{n>=1} (1 - x^(2*n)*A(x)) * (1 - x^(2*n-2)/A(x)) * (1 - x^(2*n)), due to the Jacobi triple product identity.
(4) -x*A(x)^5 = Product_{n>=1} (1 - x^(2*n)/A(x)) * (1 - x^(2*n-2)*A(x)) * (1 - x^(2*n)), due to the Jacobi triple product identity.

A357225 Coefficients in the power series A(x) such that: x*A(x)^5 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) * A(x)^n.

Original entry on oeis.org

1, 1, 6, 54, 542, 5950, 69089, 834807, 10387628, 132206325, 1713016233, 22520857313, 299667203315, 4028078782339, 54615552455056, 746073353306341, 10258385111897258, 141862903772876529, 1971827463536643265, 27532294076219156008, 386001188585539328720
Offset: 0

Views

Author

Paul D. Hanna, Sep 18 2022

Keywords

Examples

			G.f.: A(x) = 1 + x + 6*x^2 + 54*x^3 + 542*x^4 + 5950*x^5 + 69089*x^6 + 834807*x^7 + 10387628*x^8 + 132206325*x^9 + 1713016233*x^10 + ...
such that
x*A(x)^5 = ... + x^12/A(x)^4 - x^6/A(x)^3 + x^2/A(x)^2 - 1/A(x) + 1 - x^2*A(x) + x^6*A(x)^2 - x^12*A(x)^3 + x^20*A(x)^4 + ... + (-1)^n * x^(n*(n+1)) * A(x)^n + ...

Crossrefs

Cf. A355357, A357221, A357222, A357223, A357224, A357226.

Programs

  • PARI
    {a(n,p=5) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff( x*Ser(A)^p - sum(m=-ceil(sqrt(n)), ceil(sqrt(n)), (-1)^m*x^(m*(m+1))*Ser(A)^m ), #A-1)); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) satisfies:
(1) x*A(x)^5 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) * A(x)^n.
(2) -x*A(x)^6 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) / A(x)^n.
(3) x*A(x)^5 = Product_{n>=1} (1 - x^(2*n)*A(x)) * (1 - x^(2*n-2)/A(x)) * (1 - x^(2*n)), due to the Jacobi triple product identity.
(4) -x*A(x)^6 = Product_{n>=1} (1 - x^(2*n)/A(x)) * (1 - x^(2*n-2)*A(x)) * (1 - x^(2*n)), due to the Jacobi triple product identity.

A357226 Coefficients in the power series A(x) such that: x*A(x)^6 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) * A(x)^n.

Original entry on oeis.org

1, 1, 7, 73, 861, 11112, 151822, 2159143, 31627140, 473909468, 7230035454, 111924733904, 1753728878625, 27759947012294, 443247756591472, 7130680715081049, 115466397372003479, 1880525144522628300, 30783524695736369568, 506215648672559259036, 8358521379108937920413
Offset: 0

Views

Author

Paul D. Hanna, Sep 18 2022

Keywords

Examples

			G.f.: A(x) = 1 + x + 7*x^2 + 73*x^3 + 861*x^4 + 11112*x^5 + 151822*x^6 + 2159143*x^7 + 31627140*x^8 + 473909468*x^9 + 7230035454*x^10 + ...
such that
x*A(x)^6 = ... + x^12/A(x)^4 - x^6/A(x)^3 + x^2/A(x)^2 - 1/A(x) + 1 - x^2*A(x) + x^6*A(x)^2 - x^12*A(x)^3 + x^20*A(x)^4 + ... + (-1)^n * x^(n*(n+1)) * A(x)^n + ...

Crossrefs

Cf. A355357, A357221, A357222, A357223, A357224, A357225.

Programs

  • PARI
    {a(n,p=6) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff( x*Ser(A)^p - sum(m=-ceil(sqrt(n)), ceil(sqrt(n)), (-1)^m*x^(m*(m+1))*Ser(A)^m ), #A-1)); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) satisfies:
(1) x*A(x)^6 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) * A(x)^n.
(2) -x*A(x)^7 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) / A(x)^n.
(3) x*A(x)^6 = Product_{n>=1} (1 - x^(2*n)*A(x)) * (1 - x^(2*n-2)/A(x)) * (1 - x^(2*n)), due to the Jacobi triple product identity.
(4) -x*A(x)^7 = Product_{n>=1} (1 - x^(2*n)/A(x)) * (1 - x^(2*n-2)*A(x)) * (1 - x^(2*n)), due to the Jacobi triple product identity.

A359721 a(n) = coefficient of x^n in the power series A(x) such that: x = Sum_{n=-oo..+oo} (-1)^n * x^n * (1 + x^n)^n * A(x)^n.

Original entry on oeis.org

1, 1, 3, 10, 37, 127, 460, 1710, 6461, 24851, 96921, 382358, 1522997, 6116518, 24740564, 100698617, 412126133, 1694982357, 7001729420, 29037602898, 120856092153, 504647152650, 2113469775619, 8875358529059, 37364827472930, 157668052571948, 666735804080597, 2825054673048981
Offset: 0

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Author

Paul D. Hanna, Jan 11 2023

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 + 3*x^2 + 10*x^3 + 37*x^4 + 127*x^5 + 460*x^6 + 1710*x^7 + 6461*x^8 + 24851*x^9 + 96921*x^10 + 382358*x^11 + 1522997*x^12 + ...
SPECIFIC VALUES.
A(2/9) = 2.24070435506724977359903344036738515875266644317987374...
A(x) = 2 at x = 0.21791735938682393028374635435485389216073583164032813...
A(1/5) = 1.63325728843716074555468074513852677972333543319428229...
A(1/6) = 1.36828627213340815002770404510072582545059876619425902...

Links

Crossrefs

Cf. A359720, A355357, A357797, A359723, A359724.

Programs

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

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following.
(1) x = Sum_{n=-oo..+oo} (-1)^n * x^n * (1 + x^n)^n * A(x)^n.
(2) x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / ((1 + x^n)^n * A(x)^n).
(3) a(n) = Sum_{k=0..floor(2*n/3)} A359720(n,k), for n >= 0.
a(n) ~ c * d^n / n^(3/2), where d = 4.470597712126170109... and c = 1.18164918660560739... - Vaclav Kotesovec, Mar 14 2023

A359723 a(n) = coefficient of x^n in the power series A(x) such that: x = Sum_{n=-oo..+oo} (-1)^n * x^n * (3 + x^n)^n * A(x)^n.

Original entry on oeis.org

1, 1, 7, 28, 151, 803, 4108, 22532, 125449, 705929, 4035955, 23332364, 136111591, 800561116, 4741777880, 28258286033, 169322163149, 1019483819757, 6164900341534, 37425357962592, 228002416106605, 1393503512669230, 8541839907812651, 52500559705299795, 323483846045526418
Offset: 0

Views

Author

Paul D. Hanna, Jan 11 2023

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 + 7*x^2 + 28*x^3 + 151*x^4 + 803*x^5 + 4108*x^6 + 22532*x^7 + 125449*x^8 + 705929*x^9 + 4035955*x^10 + 23332364*x^11 + 136111591*x^12 + ...
SPECIFIC VALUES.
A(x) = 2 at x = 0.150684304746792807618050217238804920801612774142866...
A(1/7) = 1.67848119643298635311797131334138331526984303696733717...
A(1/8) = 1.40389487408504106142147713148599989460789630965507028...

Crossrefs

Cf. A359720, A355357, A357797, A359721, A359724.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff(x - sum(n=-#A-1, #A+1, (-1)^n * x^n * (3 + x^n +x*O(x^#A) )^n * Ser(A)^n ), #A-1) ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following.
(1) x = Sum_{n=-oo..+oo} (-1)^n * x^n * (3 + x^n)^n * A(x)^n.
(2) x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / ((1 + 3*x^n)^n * A(x)^n).
(3) a(n) = Sum_{k=0..floor(2*n/3)} A359720(n,k)*3^k, for n >= 0.

A355870 G.f. A(x,y) = Sum_{n>=0} x^n/(1-y)^(2*n+1) * Sum_{k=0..3*n} T(n,k)*y^k satisfies: y = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x,y)^n.

Original entry on oeis.org

1, 0, 3, -3, 1, 0, 9, -18, 21, -15, 6, -1, 0, 22, -56, 116, -182, 196, -140, 64, -17, 2, 0, 51, -144, 496, -1329, 2436, -3148, 2934, -1971, 934, -297, 57, -5, 0, 108, -270, 1680, -7005, 18846, -36302, 52462, -57914, 49060, -31724, 15412, -5455, 1330, -200, 14, 0, 221, -381, 5647, -32760, 116068, -298976, 591690, -920249, 1138052, -1125135, 889253, -558740, 275744, -104672, 29524, -5833, 721, -42
Offset: 0

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Author

Paul D. Hanna, Jul 19 2022

Keywords

Comments

Row sums equal A000108, the Catalan numbers:
Sum_{k=0..3*n} T(n,k) = A000108(n) for n >= 0.
T(n,3*n) = (-1)^(n-1) * A000108(n-1) for n >= 1 (Catalan numbers).
Conjecture: T(n,1) = A000716(n) for n >= 1 (number of partitions of n into parts of 3 kinds).
The generating functions of some related sequences are given as follows.
(1) A(x,x) = Sum_{n>=0} A355351(n)*x^n.
(2) A(x,2*x) = Sum_{n>=0} A355352(n)*x^n.
(3) A(x,3*x) = Sum_{n>=0} A355353(n)*x^n.
(4) A(x,4*x) = Sum_{n>=0} A355354(n)*x^n.
(5) A(x,5*x) = Sum_{n>=0} A355355(n)*x^n.
(6) A(x,x^2) = Sum_{n>=0} A355356(n)*x^n.
(7) A(x^2,x) = Sum_{n>=0} A355357(n)*x^n.
(8) A(x,x*y) = Sum_{n>=0} x^n * Sum_{k=0..n} A355350(n,k) * y^k.
(9) 1/A(4*x,-1) = 2*Sum_{n>=0} A268300(n)*x^n.
(10) A(x,2) = -Sum_{n>=0} A355871(n)*x^n.
SPECIFIC VALUES.
(V.1) A(x,y) = -exp(-Pi) at x = exp(-2*Pi) and y = exp(Pi) * Pi^(1/4)/gamma(3/4).
(V.2) A(x,y) = -exp(-2*Pi) at x = exp(-4*Pi) and y = exp(2*Pi) * Pi^(1/4)/gamma(3/4) * (6 + 4*sqrt(2))^(1/4)/2.
(V.3) A(x,y) = -exp(-3*Pi) at x = exp(-6*Pi) and y = exp(3*Pi) * Pi^(1/4)/gamma(3/4) * (27 + 18*sqrt(3))^(1/4)/3.
(V.4) A(x,y) = -exp(-4*Pi) at x = exp(-8*Pi) and y = exp(4*Pi) * Pi^(1/4)/gamma(3/4) * (8^(1/4) + 2)/4.
(V.5) A(x,y) = -exp(-sqrt(3)*Pi) at x = exp(-2*sqrt(3)*Pi) and y = exp(sqrt(3)*Pi) * gamma(4/3)^(3/2)*3^(13/8)/(Pi*2^(2/3)).

Examples

			G.f.: A(x,y) = 1/(1-y) + x*(y^3 - 3*y^2 + 3*y)/(1-y)^3 + x^2*(-y^6 + 6*y^5 - 15*y^4 + 21*y^3 - 18*y^2 + 9*y)/(1-y)^5 + x^3*(2*y^9 - 17*y^8 + 64*y^7 - 140*y^6 + 196*y^5 - 182*y^4 + 116*y^3 - 56*y^2 + 22*y)/(1-y)^7 + x^4*(-5*y^12 + 57*y^11 - 297*y^10 + 934*y^9 - 1971*y^8 + 2934*y^7 - 3148*y^6 + 2436*y^5 - 1329*y^4 + 496*y^3 - 144*y^2 + 51*y)/(1-y)^9 + x^5*(14*y^15 - 200*y^14 + 1330*y^13 - 5455*y^12 + 15412*y^11 - 31724*y^10 + 49060*y^9 - 57914*y^8 + 52462*y^7 - 36302*y^6 + 18846*y^5 - 7005*y^4 + 1680*y^3 - 270*y^2 + 108*y)/(1-y)^11 + ...
where
y = ... + x^6/A(x,y)^4 - x^3/A(x,y)^3 + x/A(x,y)^2 - 1/A(x,y) + 1 - x*A(x,y) + x^3*A(x,y)^2 - x^6*A(x,y)^3 + x^10*A(x,y)^4 -+ ... + (-1)^n * x^(n*(n+1)/2) * A(x,y)^n + ...
also,
y = (1 - x*A(x,y))*(1 - 1/A(x,y))*(1-x) * (1 - x^2*A(x,y))*(1 - x/A(x,y))*(1-x^2) * (1 - x^3*A(x,y))*(1 - x^2/A(x,y))*(1-x^3) * (1 - x^4*A(x,y))*(1 - x^3/A(x,y))*(1-x^4) * ... * (1 - x^n*A(x,y))*(1 - x^(n-1)/A(x,y))*(1-x^n) * ...
This triangle of coefficients T(n,k) of x^n*y^k/(1-y)^(2*n+1) in A(x,y), for k = 0..3*n in row n, begins
n = 0: [1];
n = 1: [0, 3, -3, 1];
n = 2: [0, 9, -18, 21, -15, 6, -1];
n = 3: [0, 22, -56, 116, -182, 196, -140, 64, -17, 2];
n = 4: [0, 51, -144, 496, -1329, 2436, -3148, 2934, -1971, 934, -297, 57, -5];
n = 5: [0, 108, -270, 1680, -7005, 18846, -36302, 52462, -57914, 49060, -31724, 15412, -5455, 1330, -200, 14];
n = 6: [0, 221, -381, 5647, -32760, 116068, -298976, 591690, -920249, 1138052, -1125135, 889253, -558740, 275744, -104672, 29524, -5833, 721, -42];
n = 7: [0, 429, -63, 18281, -134985, 594399, -1941037, 4947447, -10062669, 16571700, -22316250, 24716922, -22564425, 16956135, -10435305, 5210319, -2078910, 647565, -151825, 25215, -2646, 132]; ...
The rightmost border equals the signed Catalan numbers (A000108) shifted right one place.
Column 1 appears to equal A000716 (ignoring the initial term).
Example: at y = x, we have the g.f. of A355351:
A(x,x) = 1/(1-x) + x*(3*x - 3*x^2 + x^3)/(1-x)^3 + x^2*(9*x - 18*x^2 + 21*x^3 - 15*x^4 + 6*x^5 - x^6)/(1-x)^5 + x^3*(22*x - 56*x^2 + 116*x^3 - 182*x^4 + 196*x^5 - 140*x^6 + 64*x^7 - 17*x^8 + 2*x^9)/(1-x)^7 + ... = 1 + x + 4*x^2 + 16*x^3 + 60*x^4 + 231*x^5 + 920*x^6 + 3819*x^7 + ... + A355351(n)*x^n + ...
where x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x,x)^n.

Links

Crossrefs

Cf. A000108 (row sums), A355871 (y=2).
Cf. A355350 (related triangle), A355351 (y=x), A355352 (y=2*x), A355353 (y=3*x), A355354 (y=4*x), A355355 (y=5*x), A355356 (y=x^2), A355357 (x=x^2,y=x).
Cf. A355360 (related triangle), A000716.

Programs

  • PARI
    {T(n,k) = my(A=[1/(1-y)],t); for(i=1,n, A=concat(A,0); t = ceil(sqrt(2*(#A)+9));
A[#A] = polcoeff( (y - sum(m=-t,t, (-1)^m * x^(m*(m+1)/2) * Ser(A)^m )), #A-1,x)/(1-y)^2);polcoeff(A[n+1]*(1-y)^(2*n+1),k,y)}
for(n=0,12, for(k=0,3*n, print1( T(n,k),", "));print(""))

Formula

G.f. A(x,y) = Sum_{n>=0} x^n/(1-y)^(2*n+1) * Sum_{k=0..3*n} T(n,k)*y^k satisfies:
(1) y = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)/2) * A(x,y)^n.
(2) y = Product_{n>=1} (1 - x^n*A(x,y)) * (1 - x^(n-1)/A(x,y)) * (1 - x^n), by the Jacobi triple product identity.

A359724 a(n) = coefficient of x^n in the power series A(x) such that: x = Sum_{n=-oo..+oo} (-1)^n * x^n * (4 + x^n)^n * A(x)^n.

Original entry on oeis.org

1, 1, 9, 40, 235, 1456, 8323, 51510, 324674, 2061746, 13308492, 86876405, 572169044, 3799139674, 25403610485, 170901457100, 1155976005944, 7856772779823, 53630378512469, 367507023955203, 2527254094342404, 17435029150904202, 120633291776867632, 836907189915348056
Offset: 0

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Author

Paul D. Hanna, Jan 11 2023

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 + 9*x^2 + 40*x^3 + 235*x^4 + 1456*x^5 + 8323*x^6 + 51510*x^7 + 324674*x^8 + 2061746*x^9 + 13308492*x^10 + 86876405*x^11 + 572169044*x^12 + ...

Crossrefs

Cf. A359720, A355357, A357797, A359721, A359723.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff(x - sum(n=-#A-1, #A+1, (-1)^n * x^n * (4 + x^n +x*O(x^#A) )^n * Ser(A)^n ), #A-1) ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following.
(1) x = Sum_{n=-oo..+oo} (-1)^n * x^n * (4 + x^n)^n * A(x)^n.
(2) x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / ((1 + 4*x^n)^n * A(x)^n).
(3) a(n) = Sum_{k=0..floor(2*n/3)} A359720(n,k)*4^k, for n >= 0.

A359725 a(n) = A359720(n+2,1), for n >= 0.

Original entry on oeis.org

2, 5, 21, 51, 170, 454, 1367, 3776, 11062, 31054, 89935, 254654, 733725, 2088612, 6004175, 17150397, 49267851, 141065942, 405274932, 1162440833, 3341173303, 9596468129, 27600014912, 79359955225, 228397685542, 657335642733, 1893081845674, 5452722985712
Offset: 0

Views

Author

Paul D. Hanna, Jan 14 2023

Keywords

Comments

The g.f. of A359720, G(x,y) = Sum_{n>=0} Sum_{k=0..floor(2*n/3)} A359720(n,k)*x^n*y^k, satisfies: x = Sum_{n=-oo..+oo} (-1)^n * x^n * (y + x^n)^n * G(x,y)^n.

Crossrefs

Cf. A359720, A355357, A359726.

Programs

  • PARI
    /* a(n) = A359720(n+2,1) */
{a(n) = my(A=[1]); for(i=1, n+3, A=concat(A, 0);
A[#A] = polcoeff(x - sum(m=-#A, #A, (-1)^m * x^m * (y + x^m +x*O(x^#A) )^m * Ser(A)^m ), #A-1) );
polcoeff( polcoeff(Ser(A), n+2,x), 1,y)}
for(n=0, 30, print1(a(n), ", "))

A359726 a(n) = A359720(n+3,2), for n >= 0.

Original entry on oeis.org

1, 9, 49, 179, 711, 2390, 8361, 27082, 89389, 283170, 905307, 2825245, 8854116, 27341969, 84550769, 259046260, 793589833, 2416512240, 7352490113, 22279068811, 67435591018, 203525629398, 613550161717, 1845654390776, 5545861291941, 16637001197044, 49858191850323
Offset: 0

Views

Author

Paul D. Hanna, Jan 14 2023

Keywords

Comments

The g.f. of A359720, G(x,y) = Sum_{n>=0} Sum_{k=0..floor(2*n/3)} A359720(n,k)*x^n*y^k, satisfies: x = Sum_{n=-oo..+oo} (-1)^n * x^n * (y + x^n)^n * G(x,y)^n.

Crossrefs

Cf. A359720, A355357, A359725.

Programs

  • PARI
    /* a(n) = A359720(n+3,2) */
{a(n) = my(A=[1]); for(i=1, n+3, A=concat(A, 0);
A[#A] = polcoeff(x - sum(m=-#A, #A, (-1)^m * x^m * (y + x^m +x*O(x^#A) )^m * Ser(A)^m ), #A-1) );
polcoeff( polcoeff(Ser(A), n+3,x), 2,y)}
for(n=0, 30, print1(a(n), ", "))
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