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 1-5 of 5 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

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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)

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.

A363182 Expansion of g.f. A(x) satisfying 2 = Sum_{n=-oo..+oo} (-1)^n * x^n * (2*A(x) + x^(2*n-1))^(n+1).

Original entry on oeis.org

1, 2, 6, 20, 68, 234, 824, 2956, 10750, 39540, 146864, 550096, 2075432, 7880046, 30086704, 115445028, 444941028, 1721720032, 6686357238, 26051961396, 101810056296, 398962013908, 1567354966200, 6171824148252, 24355381522328, 96304034538898, 381506619687824
Offset: 0

Views

Author

Paul D. Hanna, May 20 2023

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 6*x^2 + 20*x^3 + 68*x^4 + 234*x^5 + 824*x^6 + 2956*x^7 + 10750*x^8 + 39540*x^9 + 146864*x^10 + ...

Links

Crossrefs

Cf. A363142, A363183, A363184, A363185.
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 * (2*Ser(A) + x^(2*m-1))^(m+1) ), #A-1)/2); 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) 2 = Sum_{n=-oo..+oo} (-1)^n * x^n * (2*A(x) + x^(2*n-1))^(n+1).
(2) 2*x = Sum_{n=-oo..+oo} (-1)^n * x^(2*n*(n-1)) / (1 + 2*A(x)*x^(2*n+1))^(n-1).
(3) A(x) = 1 / Sum_{n=-oo..+oo} (-1)^n * x^n * (2*A(x) + x^(2*n-1))^n.
(4) A(x) = x / Sum_{n=-oo..+oo} (-1)^n * x^(3*n) * (2*A(x) + x^(2*n-1))^(n-1).
(5) A(x) = 1 / Sum_{n=-oo..+oo} (-1)^n * x^(2*n^2) / (1 + 2*A(x)*x^(2*n+1))^n.
a(n) = Sum_{k=0..floor(n/2)} A359670(n-k,n-2*k) * 2^(n-2*k) for n >= 0.

A363183 Expansion of g.f. A(x) satisfying 3 = Sum_{n=-oo..+oo} (-1)^n * x^n * (3*A(x) + x^(2*n-1))^(n+1).

Original entry on oeis.org

1, 3, 11, 45, 193, 846, 3779, 17169, 79115, 368820, 1736169, 8241039, 39400672, 189567594, 917146729, 4459208292, 21776797603, 106771412718, 525382657858, 2593665077634, 12842387591191, 63762186132387, 317373771999035, 1583380006374078, 7916456438276103
Offset: 0

Views

Author

Paul D. Hanna, May 20 2023

Keywords

Examples

			G.f.: A(x) = 1 + 3*x + 11*x^2 + 45*x^3 + 193*x^4 + 846*x^5 + 3779*x^6 + 17169*x^7 + 79115*x^8 + 368820*x^9 + 1736169*x^10 + ...

Links

Crossrefs

Cf. A363142, A363182, A363184, A363185.
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 * (3*Ser(A) + x^(2*m-1))^(m+1) ), #A-1)/3); 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) 3 = Sum_{n=-oo..+oo} (-1)^n * x^n * (3*A(x) + x^(2*n-1))^(n+1).
(2) 3*x = Sum_{n=-oo..+oo} (-1)^n * x^(2*n*(n-1)) / (1 + 3*A(x)*x^(2*n+1))^(n-1).
(3) A(x) = 1 / Sum_{n=-oo..+oo} (-1)^n * x^n * (3*A(x) + x^(2*n-1))^n.
(4) A(x) = x / Sum_{n=-oo..+oo} (-1)^n * x^(3*n) * (3*A(x) + x^(2*n-1))^(n-1).
(5) A(x) = 1 / Sum_{n=-oo..+oo} (-1)^n * x^(2*n^2) / (1 + 3*A(x)*x^(2*n+1))^n.
a(n) = Sum_{k=0..floor(n/2)} A359670(n-k,n-2*k) * 3^(n-2*k) for n >= 0.

A363185 Expansion of g.f. A(x) satisfying 5 = Sum_{n=-oo..+oo} (-1)^n * x^n * (5*A(x) + x^(2*n-1))^(n+1).

Original entry on oeis.org

1, 5, 27, 155, 929, 5730, 36083, 230935, 1497739, 9822060, 65021849, 433937545, 2916359840, 19720710150, 134078691289, 915994242780, 6284957607075, 43291450899490, 299248617182754, 2075172105905550, 14432704539830007, 100648564848019045, 703624464015723819
Offset: 0

Views

Author

Paul D. Hanna, May 20 2023

Keywords

Examples

			G.f.: A(x) = 1 + 5*x + 27*x^2 + 155*x^3 + 929*x^4 + 5730*x^5 + 36083*x^6 + 230935*x^7 + 1497739*x^8 + 9822060*x^9 + 65021849*x^10 + ...

Crossrefs

Cf. A363142, A363182, A363183, A363184.
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 * (5*Ser(A) + x^(2*m-1))^(m+1) ), #A-1)/5); 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) 5 = Sum_{n=-oo..+oo} (-1)^n * x^n * (5*A(x) + x^(2*n-1))^(n+1).
(2) 5*x = Sum_{n=-oo..+oo} (-1)^n * x^(2*n*(n-1)) / (1 + 5*A(x)*x^(2*n+1))^(n-1).
(3) A(x) = 1 / Sum_{n=-oo..+oo} (-1)^n * x^n * (5*A(x) + x^(2*n-1))^n.
(4) A(x) = x / Sum_{n=-oo..+oo} (-1)^n * x^(3*n) * (5*A(x) + x^(2*n-1))^(n-1).
(5) A(x) = 1 / Sum_{n=-oo..+oo} (-1)^n * x^(2*n^2) / (1 + 5*A(x)*x^(2*n+1))^n.
a(n) = Sum_{k=0..floor(n/2)} A359670(n-k,n-2*k) * 5^(n-2*k) for n >= 0.
Showing 1-5 of 5 results.

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