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 1-10 of 10 results.

A359670 Triangle of coefficients T(n,k) of x^n*y^k in g.f. A(x,y) satisfying y = Sum_{n=-oo..+oo} (-1)^n * x^n * (y*A(x,y) + x^(n-1))^(n+1).

Original entry on oeis.org

1, 2, 1, 4, 6, 1, 8, 21, 12, 1, 14, 62, 68, 20, 1, 24, 162, 284, 170, 30, 1, 40, 384, 998, 970, 360, 42, 1, 64, 855, 3092, 4410, 2720, 679, 56, 1, 100, 1806, 8724, 17172, 15627, 6608, 1176, 72, 1, 154, 3648, 22904, 59545, 74682, 47089, 14392, 1908, 90, 1, 232, 7110, 56679, 188700, 311530, 271698, 125160, 28764, 2940, 110, 1
Offset: 0

Views

Author

Paul D. Hanna, Jan 17 2023

Keywords

Comments

Related identity: 0 = Sum_{-oo..+oo} (-1)^n * x^n * (y + x^n)^n, which holds formally for all y.
T(n,0) = A015128(n), the number of overpartitions of n, for n >= 0.
T(n+1,1) = A022571(n), the coefficient of x^n in Product_{m>=1} (1 + x^m)^6, for n >= 0.
A359711(n) = Sum_{k=0..n} T(n,k) for n >= 0 (row sums).
A359712(n) = Sum_{k=0..n} T(n,k)*2^k for n >= 0.
A359713(n) = Sum_{k=0..n} T(n,k)*3^k for n >= 0.
A363104(n) = Sum_{k=0..n} T(n,k)*4^k for n >= 0.
A363105(n) = Sum_{k=0..n} T(n,k)*5^k for n >= 0.
A359714(n) = T(2*n,n) for n >= 0 (central terms).
A359715(n) = T(n+2,2) for n >= 0.
A359718(n) = T(n+3,3) for n >= 0.
A363142(n) = Sum_{k=0..floor(n/2)} T(n-k,n-2*k) for n >= 0. - Paul D. Hanna, May 18 2023
From Paul D. Hanna, May 20 2023: (Start)
A363182(n) = Sum_{k=0..floor(n/2)} T(n-k,n-2*k) * 2^(n-2*k) for n >= 0.
A363183(n) = Sum_{k=0..floor(n/2)} T(n-k,n-2*k) * 3^(n-2*k) for n >= 0.
A363184(n) = Sum_{k=0..floor(n/2)} T(n-k,n-2*k) * 4^(n-2*k) for n >= 0.
A363185(n) = Sum_{k=0..floor(n/2)} T(n-k,n-2*k) * 5^(n-2*k) for n >= 0. (End)

Examples

			G.f.: A(x,y) = 1 + x*(2 + y) + x^2*(4 + 6*y + y^2) + x^3*(8 + 21*y + 12*y^2 + y^3) + x^4*(14 + 62*y + 68*y^2 + 20*y^3 + y^4) + x^5*(24 + 162*y + 284*y^2 + 170*y^3 + 30*y^4 + y^5) + x^6*(40 + 384*y + 998*y^2 + 970*y^3 + 360*y^4 + 42*y^5 + y^6) + x^7*(64 + 855*y + 3092*y^2 + 4410*y^3 + 2720*y^4 + 679*y^5 + 56*y^6 + y^7) + x^8*(100 + 1806*y + 8724*y^2 + 17172*y^3 + 15627*y^4 + 6608*y^5 + 1176*y^6 + 72*y^7 + y^8) + x^9*(154 + 3648*y + 22904*y^2 + 59545*y^3 + 74682*y^4 + 47089*y^5 + 14392*y^6 + 1908*y^7 + 90*y^8 + y^9) + x^10*(232 + 7110*y + 56679*y^2 + 188700*y^3 + 311530*y^4 + 271698*y^5 + 125160*y^6 + 28764*y^7 + 2940*y^8 + 110*y^9 + y^10) + ...
This triangle of coefficients T(n,k) of x^n*y^k in A(x,y), for n >= 0, k = 0..n, begins
[1];
[2, 1];
[4, 6, 1];
[8, 21, 12, 1];
[14, 62, 68, 20, 1];
[24, 162, 284, 170, 30, 1];
[40, 384, 998, 970, 360, 42, 1];
[64, 855, 3092, 4410, 2720, 679, 56, 1];
[100, 1806, 8724, 17172, 15627, 6608, 1176, 72, 1];
[154, 3648, 22904, 59545, 74682, 47089, 14392, 1908, 90, 1];
[232, 7110, 56679, 188700, 311530, 271698, 125160, 28764, 2940, 110, 1];
[344, 13434, 133516, 556085, 1169100, 1342684, 860664, 300888, 53640, 4345, 132, 1];
[504, 24702, 301664, 1542640, 4029237, 5884160, 4980320, 2438712, 666240, 94490, 6204, 156, 1];
[728, 44361, 657368, 4065868, 12940766, 23411339, 25215416, 16367874, 6302148, 1377464, 158708, 8606, 182, 1];
[1040, 78006, 1387854, 10253720, 39153924, 85994062, 114672768, 94919382, 48660900, 15071628, 2687454, 256022, 11648, 210, 1]; ...
RELATED SERIES.
Given g.f. F(x) of A361770, where
F(x) = 1 + 3*x + 14*x^2 + 80*x^3 + 510*x^4 + 3498*x^5 + 25145*x^6 + 186972*x^7 + 1426159*x^8 + 11096944*x^9 + 87736474*x^10 + ... + A361770(n)*x^n + ...
then
(1) F(x) = Sum_{n>=0} x^n * Sum_{k=0..n} T(n,k) * F(x)^k,
(2) F(x) = Sum_{n=-oo..+oo} (-1)^n * x^n * (F(x)^2 + x^(n-1))^(n+1).
Given g.f. G(x) of A363135, where
G(x) = 1 + 3*x + 17*x^2 + 133*x^3 + 1201*x^4 + 11796*x^5 + 122192*x^6 + 1314266*x^7 + 14536760*x^8 + 164299909*x^9 + ... + A363135(n)*x^n + ...
then
(1) G(x) = Sum_{n>=0} x^n * Sum_{k=0..n} T(n,k) * G(x)^(2*k),
(2) G(x)^2 = Sum_{n=-oo..+oo} (-1)^n * x^n * (F(x)^3 + x^(n-1))^(n+1).

Links

Crossrefs

Cf. A359711 (row sums), A359712 (y=2), A359713 (y=3), A363104(y=4), A363105 (y=5).
Cf. A359714 (central terms), A359715 (column 2), A359718 (column 3).
Cf. A363142, A363182, A363183, A363184, A363185.
Cf. A361770, A363135, A363136, A363137.
Cf. A359720, A293600.

Programs

  • PARI
    {T(n,k) = my(A=1); for(i=1,n,
A = 1/sum(m=-#A,#A, (-1)^m * (x*y*A + x^m + x*O(x^n) )^m ) );
polcoeff( polcoeff( A,n,x),k,y)}
for(n=0,15, for(k=0,n, print1( T(n,k),", "));print(""))
  • PARI
    {T(n,k) = my(A=[1]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff(-y + sum(n=-#A,#A, (-1)^n * x^n * (y*Ser(A) + x^(n-1))^(n+1) )/(-y),#A-1,x) ); polcoeff( A[n+1],k,y)}
for(n=0,15, for(k=0,n, print1( T(n,k),", "));print(""))

Formula

G.f. A(x,y) = Sum_{n>=0} Sum_{k=0..n} T(n,k) * x^n*y^k may be described as follows.
(1) y = Sum_{n=-oo..+oo} (-1)^n * x^n * (y*A(x,y) + x^(n-1))^(n+1).
(2) x*y = Sum_{n=-oo..+oo} (-1)^n * (x*y*A(x,y) + x^n)^(n+1).
(3) x*y = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + y*A(x,y)*x^(n+1))^(n-1).
(4) A(x,y) = 1/[Sum_{n=-oo..+oo} (-1)^n * (x*y*A(x,y) + x^n)^n ].
(5) A(x,y) = 1/[Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + y*A(x,y)*x^(n+1))^n ].
From Paul D. Hanna, May 18 2023: (Start)
(6) y = Sum_{n=-oo..+oo} (-1)^n * x^(3*n+1) * (y*A(x,y) + x^n)^n.
(7) A(x,y) = 1/[Sum_{n=-oo..+oo} (-1)^(n+1) * x^(2*n+1) * (y*A(x,y) + x^n)^n ].
(8) x*y = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(n*(n-1)) / (1 + y*A(x,y)*x^(n+1))^(n+1).
(9) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n) * (y*A(x,y) + x^n)^(n+1).
(10) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + y*A(x,y)*x^n)^n.
(11) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + y*A(x,y)*x^(n+1))^n. (End)

A359712 a(n) = coefficient of x^n in A(x) such that 2 = Sum_{n=-oo..+oo} (-x)^n * (2*A(x) + x^(n-1))^(n+1).

Original entry on oeis.org

1, 4, 20, 106, 586, 3356, 19728, 118382, 722208, 4466050, 27931600, 176371300, 1122867012, 7199842666, 46454345844, 301384205640, 1964899532794, 12866563846920, 84585757496444, 558060746899684, 3693810227983576, 24521903234307786, 163234951757526400
Offset: 0

Views

Author

Paul D. Hanna, Jan 17 2023

Keywords

Examples

			G.f.: A(x) = 1 + 4*x + 20*x^2 + 106*x^3 + 586*x^4 + 3356*x^5 + 19728*x^6 + 118382*x^7 + 722208*x^8 + 4466050*x^9 + 27931600*x^10 +  ...

Links

Crossrefs

Cf. A359670, A359711, A359713, A363104, A363105, A361778.

Programs

  • PARI
    {a(n) = my(A=1,y=2); for(i=1,n,
A = 1/sum(m=-#A,#A, (-1)^m * (x*y*A + x^m + x*O(x^n) )^m ) );
polcoeff( A,n,x)}
for(n=0,25, print1( a(n),", "))
  • PARI
    {a(n) = my(A=[1],y=2); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff(-y + sum(n=-#A,#A, (-1)^n * x^n * (y*Ser(A) + x^(n-1))^(n+1) )/(-y),#A-1,x) ); A[n+1]}
for(n=0,25, print1( a(n),", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n) * x^n may be described as follows.
(1) 2 = Sum_{n=-oo..+oo} (-1)^n * x^n * (2*A(x) + x^(n-1))^(n+1).
(2) 2*x = Sum_{n=-oo..+oo} (-1)^n * (2*x*A(x) + x^n)^(n+1).
(3) 2*x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + 2*A(x)*x^(n+1))^(n-1).
(4) A(x) = 1 / [Sum_{n=-oo..+oo} (-1)^n * (2*x*A(x) + x^n)^n ].
(5) A(x) = 1 / [Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + 2*A(x)*x^(n+1))^n ].
From Paul D. Hanna, May 12 2023: (Start)
(6) 2 = Sum_{n=-oo..+oo} (-1)^n * x^(3*n+1) * (2*A(x) + x^n)^n.
(7) A(x) = 1 / [Sum_{n=-oo..+oo} (-1)^(n+1) * x^(2*n+1) * (2*A(x) + x^n)^n ].
(8) 2*x = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(n*(n-1)) / (1 + 2*A(x)*x^(n+1))^(n+1).
(9) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n) * (2*A(x) + x^n)^(n+1).
(10) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + 2*A(x)*x^n)^n.
(11) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + 2*A(x)*x^(n+1))^n. (End)
a(n) = Sum_{k=0..n} A359670(n,k)*2^k for n >= 0.

A363142 Expansion of g.f. A(x) satisfying 1 = Sum_{n=-oo..+oo} (-1)^n * x^n * (A(x) + x^(2*n-1))^(n+1).

Original entry on oeis.org

1, 1, 3, 7, 17, 42, 107, 275, 715, 1884, 5009, 13421, 36224, 98382, 268657, 737244, 2032035, 5622938, 15615186, 43505382, 121570407, 340639265, 956861955, 2694064938, 7601455079, 21490621769, 60870280259, 172707869088, 490818655346, 1396973741672, 3981748142925
Offset: 0

Views

Author

Paul D. Hanna, May 17 2023

Keywords

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 7*x^3 + 17*x^4 + 42*x^5 + 107*x^6 + 275*x^7 + 715*x^8 + 1884*x^9 + 5009*x^10 + 13421*x^11 + 36224*x^12 + ...

Links

Crossrefs

Cf. A359711, A363143, A363144, A363182.
Cf. A359670.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff(1 - sum(m=-#A, #A, (-1)^m * x^m * (Ser(A) + x^(2*m-1))^(m+1) ),#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) 1 = Sum_{n=-oo..+oo} (-1)^n * x^n * (A(x) + x^(2*n-1))^(n+1).
(2) x = Sum_{n=-oo..+oo} (-1)^n * x^(2*n*(n-1)) / (1 + A(x)*x^(2*n+1))^(n-1).
(3) A(x) = 1 / Sum_{n=-oo..+oo} (-1)^n * x^n * (A(x) + x^(2*n-1))^n.
(4) A(x) = x / Sum_{n=-oo..+oo} (-1)^n * x^(3*n) * (A(x) + x^(2*n-1))^(n-1).
(5) A(x) = 1 / Sum_{n=-oo..+oo} (-1)^n * x^(2*n^2) / (1 + A(x)*x^(2*n+1))^n.
a(n) = Sum_{k=0..floor(n/2)} A359670(n-k,n-2*k) for n >= 0. - Paul D. Hanna, May 18 2023

A359713 a(n) = coefficient of x^n in A(x) such that 3 = Sum_{n=-oo..+oo} (-x)^n * (3*A(x) + x^(n-1))^(n+1).

Original entry on oeis.org

1, 5, 31, 206, 1433, 10329, 76459, 577855, 4440538, 34591555, 272545144, 2168118299, 17390330046, 140486973983, 1142036572271, 9335129425718, 76681549612006, 632655728172281, 5240339959916895, 43561574812700958, 363294379940353624, 3038799803831856805
Offset: 0

Views

Author

Paul D. Hanna, Jan 17 2023

Keywords

Examples

			G.f.: A(x) = 1 + 5*x + 31*x^2 + 206*x^3 + 1433*x^4 + 10329*x^5 + 76459*x^6 + 577855*x^7 + 4440538*x^8 + 34591555*x^9 + 272545144*x^10 + ...

Links

Crossrefs

Cf. A359670, A359711, A359712, A363104, A363105.

Programs

  • PARI
    {a(n) = my(A=1,y=3); for(i=1,n,
A = 1/sum(m=-#A,#A, (-1)^m * (x*y*A + x^m + x*O(x^n) )^m ) );
polcoeff( A,n,x)}
for(n=0,25, print1( a(n),", "))
  • PARI
    {a(n) = my(A=[1],y=3); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff(-y + sum(n=-#A,#A, (-1)^n * x^n * (y*Ser(A) + x^(n-1))^(n+1) )/(-y),#A-1,x) ); A[n+1]}
for(n=0,25, print1( a(n),", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n) * x^n may be described as follows.
(1) 3 = Sum_{n=-oo..+oo} (-1)^n * x^n * (3*A(x) + x^(n-1))^(n+1).
(2) 3*x = Sum_{n=-oo..+oo} (-1)^n * (3*x*A(x) + x^n)^(n+1).
(3) 3*x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + 3*A(x)*x^(n+1))^(n-1).
(4) A(x) = 1 / [Sum_{n=-oo..+oo} (-1)^n * (3*x*A(x) + x^n)^n ].
(5) A(x) = 1 / [Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + 3*A(x)*x^(n+1))^n ].
a(n) = Sum_{k=0..n} A359670(n,k)*3^k for n >= 0.

A363104 Expansion of g.f. A(x) satisfying 4 = Sum_{n=-oo..+oo} (-x)^n * (4*A(x) + x^(n-1))^(n+1).

Original entry on oeis.org

1, 6, 44, 348, 2886, 24800, 218888, 1972572, 18075100, 167900506, 1577467760, 14963979584, 143124912880, 1378756186748, 13365212659144, 130274948580864, 1276075285222662, 12554452588117632, 124003727286837484, 1229203475053859456, 12224294019862383720
Offset: 0

Views

Author

Paul D. Hanna, May 21 2023

Keywords

Comments

Conjecture: g.f. A(x) == theta_3(x) (mod 4); a(n) == 2 (mod 4) iff n is a nonzero square and a(n) == 0 (mod 4) iff n is nonsquare.

Examples

			G.f.: A(x) = 1 + 6*x + 44*x^2 + 348*x^3 + 2886*x^4 + 24800*x^5 + 218888*x^6 + 1972572*x^7 + 18075100*x^8 + 167900506*x^9 + 1577467760*x^10 + ...

Links

Crossrefs

Cf. A359670, A359711, A359712, A359713, A363105.
Cf. A363184.

Programs

  • PARI
    {a(n) = my(A=1, y=4); for(i=1, n,
A = 1/sum(m=-#A, #A, (-1)^m * (x*y*A + x^m + x*O(x^n) )^m ) );
polcoeff( A, n, x)}
for(n=0, 25, print1( a(n), ", "))
  • PARI
    {a(n) = my(A=[1], y=4); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff(-y + sum(n=-#A, #A, (-1)^n * x^n * (y*Ser(A) + x^(n-1))^(n+1) )/(-y), #A-1, x) ); A[n+1]}
for(n=0, 25, print1( a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n) * x^n may be described as follows.
(1) 4 = Sum_{n=-oo..+oo} (-1)^n * x^n * (4*A(x) + x^(n-1))^(n+1).
(2) 4 = Sum_{n=-oo..+oo} (-1)^n * x^(3*n+1) * (4*A(x) + x^n)^n.
(3) 4*x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + 4*A(x)*x^(n+1))^(n-1).
(4) 4*x = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(n*(n-1)) / (1 + 4*A(x)*x^(n+1))^(n+1).
(5) A(x) = 1 / [Sum_{n=-oo..+oo} (-1)^n * x^n * (4*A(x) + x^(n-1))^n ].
(6) A(x) = 1 / [Sum_{n=-oo..+oo} (-1)^(n+1) * x^(2*n+1) * (4*A(x) + x^n)^n ].
(7) A(x) = 1 / [Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + 4*A(x)*x^(n+1))^n ].
(8) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n) * (4*A(x) + x^n)^(n+1).
(9) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + 4*A(x)*x^n)^n.
(10) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + 4*A(x)*x^(n+1))^n.
a(n) = Sum_{k=0..n} A359670(n,k) * 4^k for n >= 0.

A363105 Expansion of g.f. A(x) satisfying 5 = Sum_{n=-oo..+oo} (-x)^n * (5*A(x) + x^(n-1))^(n+1).

Original entry on oeis.org

1, 7, 59, 538, 5149, 51059, 520035, 5407889, 57181230, 612910369, 6644662132, 72731584789, 802696690614, 8922392225233, 99798739026795, 1122441028044882, 12686176392341722, 144013323190860339, 1641303449002365323, 18772674107796041770, 215413772477355781876
Offset: 0

Views

Author

Paul D. Hanna, May 21 2023

Keywords

Examples

			G.f.: A(x) = 1 + 7*x + 59*x^2 + 538*x^3 + 5149*x^4 + 51059*x^5 + 520035*x^6 + 5407889*x^7 + 57181230*x^8 + 612910369*x^9 + 6644662132*x^10 + ...

Links

Crossrefs

Cf. A359670, A359711, A359712, A359713, A363104.
Cf. A363185.

Programs

  • PARI
    {a(n) = my(A=1, y=5); for(i=1, n,
A = 1/sum(m=-#A, #A, (-1)^m * (x*y*A + x^m + x*O(x^n) )^m ) );
polcoeff( A, n, x)}
for(n=0, 25, print1( a(n), ", "))
  • PARI
    {a(n) = my(A=[1], y=5); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff(-y + sum(n=-#A, #A, (-1)^n * x^n * (y*Ser(A) + x^(n-1))^(n+1) )/(-y), #A-1, x) ); A[n+1]}
for(n=0, 25, print1( a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n) * x^n may be described as follows.
(1) 5 = Sum_{n=-oo..+oo} (-1)^n * x^n * (5*A(x) + x^(n-1))^(n+1).
(2) 5 = Sum_{n=-oo..+oo} (-1)^n * x^(3*n+1) * (5*A(x) + x^n)^n.
(3) 5*x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + 5*A(x)*x^(n+1))^(n-1).
(4) 5*x = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(n*(n-1)) / (1 + 5*A(x)*x^(n+1))^(n+1).
(5) A(x) = 1 / [Sum_{n=-oo..+oo} (-1)^n * x^n * (5*A(x) + x^(n-1))^n ].
(6) A(x) = 1 / [Sum_{n=-oo..+oo} (-1)^(n+1) * x^(2*n+1) * (5*A(x) + x^n)^n ].
(7) A(x) = 1 / [Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + 5*A(x)*x^(n+1))^n ].
(8) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n) * (5*A(x) + x^n)^(n+1).
(9) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + 5*A(x)*x^n)^n.
(10) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + 5*A(x)*x^(n+1))^n.
a(n) = Sum_{k=0..n} A359670(n,k) * 5^k for n >= 0.

A361770 Expansion of g.f. A(x) satisfying A(x) = Sum_{n=-oo..+oo} (-1)^n * x^n * (A(x)^2 + x^(n-1))^(n+1).

Original entry on oeis.org

1, 3, 14, 80, 510, 3498, 25145, 186972, 1426159, 11096944, 87736474, 702837098, 5692337206, 46533458472, 383450469145, 3181746494524, 26562082580277, 222941953595054, 1880174585677589, 15924467403391355, 135396623401761765, 1155230973031795808, 9888061401816818319
Offset: 0

Views

Author

Paul D. Hanna, May 24 2023

Keywords

Comments

Given g.f. G(x,y) of triangle A359670, then A(x) = G(x,y=A(x)).

Examples

			G.f.: A(x) = 1 + 3*x + 14*x^2 + 80*x^3 + 510*x^4 + 3498*x^5 + 25145*x^6 + 186972*x^7 + 1426159*x^8 + 11096944*x^9 + 87736474*x^10 + ...
where A = A(x) may be generated from triangle A359670 as follows:
A(x) = 1 + x*(2 + A) + x^2*(4 + 6*A + A^2) + x^3*(8 + 21*A + 12*A^2 + A^3) + x^4*(14 + 62*A + 68*A^2 + 20*A^3 + A^4) + x^5*(24 + 162*A + 284*A^2 + 170*A^3 + 30*A^4 + A^5) + x^6*(40 + 384*A + 998*A^2 + 970*A^3 + 360*A^4 + 42*A^5 + A^6) + x^7*(64 + 855*A + 3092*A^2 + 4410*A^3 + 2720*A^4 + 679*A^5 + 56*A^6 + A^7) + x^8*(100 + 1806*A + 8724*A^2 + 17172*A^3 + 15627*A^4 + 6608*A^5 + 1176*A^6 + 72*A^7 + A^8) + ... + x^n*(Sum_{k=0..n} A359670(n,k)*A(x)^k) + ...
Also, A(x) = G(x,y=1) where G(x,y) satisfies
y*G(x,y) = Sum_{n=-oo..+oo} (-1)^n * x^n * (y*G(x,y)^2 + x^(n-1))^(n+1).
Explicitly,
G(x,y) = 1 + x*(2 + y) + x^2*(4 + 8*y + 2*y^2) + x^3*(8 + 37*y + 30*y^2 + 5*y^3) + x^4*(14 + 136*y + 234*y^2 + 112*y^3 + 14*y^4) + x^5*(24 + 432*y + 1320*y^2 + 1260*y^3 + 420*y^4 + 42*y^5) + x^6*(40 + 1232*y + 6093*y^2 + 9824*y^3 + 6240*y^4 + 1584*y^5 + 132*y^6) + x^7*(64 + 3245*y + 24402*y^2 + 60543*y^3 + 62880*y^4 + 29403*y^5 + 6006*y^6 + 429*y^7) + x^8*(100 + 8024*y + 87754*y^2 + 315616*y^3 + 490405*y^4 + 365816*y^5 + 134134*y^6 + 22880*y^7 + 1430*y^8) + x^9*(154 + 18832*y + 289812*y^2 + 1448744*y^3 + 3178302*y^4 + 3476418*y^5 + 1993992*y^6 + 598312*y^7 + 87516*y^8 + 4862*y^9) + ...

Links

Crossrefs

Cf. A359670, A359711, A363135, A363136, A363137.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff(1 - sum(n=-#A, #A, (-1)^n * x^n * (Ser(A)^2 + x^(n-1))^(n+1) )/Ser(A), #A-1, x) ); A[n+1]}
for(n=0, 25, print1( a(n), ", "))
  • PARI
    {a(n) = my(A=1); for(i=1, n,
A = 1/sum(m=-#A, #A, (-1)^m * (x*A^2 + x^m + x*O(x^n) )^m ) );
polcoeff( A, n, x)}
for(n=0, 25, print1( a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n) * x^n may be described as follows.
(1) A(x) = Sum_{n>=0} x^n * Sum_{k=0..n} A359670(n,k) * A(x)^k.
(2) A(x) = Sum_{n=-oo..+oo} (-1)^n * x^n * (A(x)^2 + x^(n-1))^(n+1).
(3) A(x) = Sum_{n=-oo..+oo} (-1)^n * x^(3*n+1) * (A(x)^2 + x^n)^n.
(4) x*A(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + A(x)^2*x^(n+1))^(n-1).
(5) x*A(x) = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(n*(n-1)) / (1 + A(x)^2*x^(n+1))^(n+1).
(6) A(x) = 1 / [Sum_{n=-oo..+oo} (-1)^n * x^n * (A(x)^2 + x^(n-1))^n ].
(7) A(x) = 1 / [Sum_{n=-oo..+oo} (-1)^(n+1) * x^(2*n+1) * (A(x)^2 + x^n)^n ].
(8) A(x) = 1 / [Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + A(x)^2*x^(n+1))^n ].
(9) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n) * (A(x)^2 + x^n)^(n+1).
(10) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + A(x)^2*x^n)^n.
(11) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + A(x)^2*x^(n+1))^n.
a(n) ~ c * d^n / n^(3/2), where d = 9.156930044633747979075094492861543774480990540... and c = 0.74413616954012053890115400925213042708811... - Vaclav Kotesovec, Jul 03 2025

A363106 Expansion of g.f. A(x) satisfying 1 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n) * (A(x) + x^(n-2))^(n+1).

Original entry on oeis.org

1, 2, 5, 14, 36, 98, 271, 752, 2124, 6052, 17375, 50292, 146469, 428992, 1262946, 3734748, 11089366, 33048498, 98819841, 296388284, 891436452, 2688029716, 8124678435, 24611028218, 74702698749, 227177047220, 692084278902, 2111883982538, 6454350205098, 19754469483978
Offset: 0

Views

Author

Paul D. Hanna, May 24 2023

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 5*x^2 + 14*x^3 + 36*x^4 + 98*x^5 + 271*x^6 + 752*x^7 + 2124*x^8 + 6052*x^9 + 17375*x^10 + 50292*x^11 + 146469*x^12 + ...

Links

Crossrefs

Cf. A359711, A363107, A363108, A363109, A363140.

Programs

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

Formula

G.f. A(x) = Sum_{n>=0} a(n) * x^n may be described as follows.
(1) 1 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n) * (A(x) + x^(n-2))^(n+1).
(2) 1 = Sum_{n=-oo..+oo} (-1)^n * x^(4*n) * (A(x) + x^(n-1))^n.
(3) x^2 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + A(x)*x^(n+2))^(n-1).
(4) x^2 = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(n*(n-1)) / (1 + A(x)*x^(n+2))^(n+1).
(5) A(x) = 1 / Sum_{n=-oo..+oo} (-1)^n * x^(2*n) * (A(x) + x^(n-2))^n.
(6) A(x) = 1 / Sum_{n=-oo..+oo} (-1)^n * x^(3*n-2) * (A(x) + x^(n-2))^(n-1).
(7) A(x) = 1 / Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + A(x)*x^(n+2))^(n+1).
(8) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n) * (A(x) + x^(n-1))^n.
(9) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + A(x)*x^(n+2))^n.

A363143 Expansion of g.f. A(x) satisfying 1 = Sum_{n=-oo..+oo} (-1)^n * x^n * (A(x) + x^(3*n-1))^(n+1).

Original entry on oeis.org

1, 1, 1, 3, 7, 13, 25, 52, 111, 235, 495, 1054, 2271, 4923, 10703, 23354, 51190, 112668, 248783, 550875, 1223107, 2722766, 6075619, 13586390, 30442616, 68339788, 153683822, 346173172, 780948750, 1764312745, 3991321375, 9040912764, 20503640896, 46552634034
Offset: 0

Views

Author

Paul D. Hanna, May 17 2023

Keywords

Examples

			G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 7*x^4 + 13*x^5 + 25*x^6 + 52*x^7 + 111*x^8 + 235*x^9 + 495*x^10 + 1054*x^11 + 2271*x^12 + ...

Links

Crossrefs

Cf. A359711, A363142, A363144.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff(1 - sum(m=-#A, #A, (-1)^m * x^m * (Ser(A) + x^(3*m-1))^(m+1) ),#A-1));A[n+1]}
for(n=0,40,print1(a(n),", "))

Formula

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

A363144 Expansion of g.f. A(x) satisfying 1 = Sum_{n=-oo..+oo} (-1)^n * x^n * (A(x) + x^(4*n-1))^(n+1).

Original entry on oeis.org

1, 1, 1, 1, 3, 7, 13, 21, 35, 64, 125, 243, 459, 852, 1593, 3035, 5857, 11326, 21835, 42053, 81246, 157741, 307421, 600207, 1172805, 2294197, 4495735, 8827574, 17363422, 34198201, 67429181, 133097669, 263028031, 520406201, 1030749582, 2043553947, 4055171751
Offset: 0

Views

Author

Paul D. Hanna, May 17 2023

Keywords

Examples

			G.f.: A(x) = 1 + x + x^2 + x^3 + 3*x^4 + 7*x^5 + 13*x^6 + 21*x^7 + 35*x^8 + 64*x^9 + 125*x^10 + 243*x^11 + 459*x^12 + ...

Links

Crossrefs

Cf. A359711, A363142, A363143.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1,n, A = concat(A,0);
A[#A] = polcoeff(1 - sum(m=-#A, #A, (-1)^m * x^m * (Ser(A) + x^(4*m-1))^(m+1) ),#A-1));A[n+1]}
for(n=0,40,print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following.
(1) 1 = Sum_{n=-oo..+oo} (-1)^n * x^n * (A(x) + x^(4*n-1))^(n+1).
(2) x = Sum_{n=-oo..+oo} (-1)^n * x^(4*n*(n-1)) / (1 + A(x)*x^(4*n+1))^(n-1).
(3) A(x) = 1 / Sum_{n=-oo..+oo} (-1)^n * x^n * (A(x) + x^(4*n-1))^n.
(4) A(x) = x / Sum_{n=-oo..+oo} (-1)^n * x^(5*n) * (A(x) + x^(4*n-1))^(n-1).
(5) A(x) = 1 / Sum_{n=-oo..+oo} (-1)^n * x^(4*n^2) / (1 + A(x)*x^(4*n+1))^n.
Showing 1-10 of 10 results.

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