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.

A359720 T(n,k) = coefficient of x^n*y^k in A(x,y) such that: x = Sum_{n=-oo..+oo} (-1)^n * x^n * (y + x^n)^n * A(x,y)^n.

Original entry on oeis.org

1, 1, 1, 2, 4, 5, 1, 7, 21, 9, 20, 51, 49, 7, 43, 170, 179, 66, 2, 110, 454, 711, 381, 54, 262, 1367, 2390, 1894, 523, 25, 674, 3776, 8361, 8070, 3496, 469, 5, 1684, 11062, 27082, 33093, 19129, 4602, 269, 4397, 31054, 89389, 125983, 93908, 33211, 4325, 91, 11320, 89935, 283170, 470439, 421762, 200449, 43062, 2846, 14
Offset: 0

Views

Author

Paul D. Hanna, Jan 13 2023

Keywords

Comments

Related identity: 0 = Sum_{n=-oo..+oo} x^n * (y - x^n)^n, which holds formally for all y.
The terms in row n start with index k = 0 to k = floor(2*n/3), for n >= 0.
A359721(n) = Sum_{k=0..floor(2*n/3)} T(n,k), for n >= 0 (row sums).
A357797(n) = Sum_{k=0..floor(2*n/3)} T(n,k)*2^k, for n >= 0.
A359723(n) = Sum_{k=0..floor(2*n/3)} T(n,k)*3^k, for n >= 0.
A359724(n) = Sum_{k=0..floor(2*n/3)} T(n,k)*4^k, for n >= 0.
A355357(n) = T(n,0), for n >= 0.
A359725(n) = T(n+2,1), for n >= 0.
A359726(n) = T(n+3,2), for n >= 0.
A000108(n) = T(3*n,2*n), for n >= 0.
A359722(n) = T(3*n+1,2*n), for n >= 0.
A097613(n+2) = T(3*n+2,2*n+1), for n >= 0.

Examples

			G.f.: A(x,y) = 1 + x*(1) + x^2*(1 + 2*y) + x^3*(4 + 5*y + y^2) + x^4*(7 + 21*y + 9*y^2) + x^5*(20 + 51*y + 49*y^2 + 7*y^3) + x^6*(43 + 170*y + 179*y^2 + 66*y^3 + 2*y^4) + x^7*(110 + 454*y + 711*y^2 + 381*y^3 + 54*y^4) + x^8*(262 + 1367*y + 2390*y^2 + 1894*y^3 + 523*y^4 + 25*y^5) + x^9*(674 + 3776*y + 8361*y^2 + 8070*y^3 + 3496*y^4 + 469*y^5 + 5*y^6) + x^10*(1684 + 11062*y + 27082*y^2 + 33093*y^3 + 19129*y^4 + 4602*y^5 + 269*y^6) + x^11*(4397 + 31054*y + 89389*y^2 + 125983*y^3 + 93908*y^4 + 33211*y^5 + 4325*y^6 + 91*y^7) + x^12*(11320 + 89935*y + 283170*y^2 + 470439*y^3 + 421762*y^4 + 200449*y^5 + 43062*y^6 + 2846*y^7 + 14*y^8) + x^13*(29938 + 254654*y + 905307*y^2 + 1683683*y^3 + 1798279*y^4 + 1072012*y^5 + 329533*y^6 + 41858*y^7 + 1254*y^8) + x^14*(78641 + 733725*y + 2825245*y^2 + 5954300*y^3 + 7287245*y^4 + 5277807*y^5 + 2131517*y^6 + 421554*y^7 + 30194*y^8 + 336*y^9 ) + x^15*(210044 + 2088612*y + 8854116*y^2 + 20499318*y^3 + 28639206*y^4 + 24326336*y^5 + 12274991*y^6 + 3370105*y^7 + 420102*y^8 + 15745*y^9 + 42*y^10) + ...
This irregular triangle of coefficients T(n,k) of x^n*y^k, for n >= 0, k = 0..[2*n/3], in g.f. A(x,y) begin:
n = 0: [1],
n = 1: [1],
n = 2: [1, 2],
n = 3: [4, 5, 1],
n = 4: [7, 21, 9],
n = 5: [20, 51, 49, 7],
n = 6: [43, 170, 179, 66, 2],
n = 7: [110, 454, 711, 381, 54],
n = 8: [262, 1367, 2390, 1894, 523, 25],
n = 9: [674, 3776, 8361, 8070, 3496, 469, 5],
n = 10: [1684, 11062, 27082, 33093, 19129, 4602, 269],
n = 11: [4397, 31054, 89389, 125983, 93908, 33211, 4325, 91],
n = 12: [11320, 89935, 283170, 470439, 421762, 200449, 43062, 2846, 14],
n = 13: [29938, 254654, 905307, 1683683, 1798279, 1072012, 329533, 41858, 1254],
n = 14: [78641, 733725, 2825245, 5954300, 7287245, 5277807, 2131517, 421554, 30194, 336],
n = 15: [210044, 2088612, 8854116, 20499318, 28639206, 24326336, 12274991, 3370105, 420102, 15745, 42], ...
...
in which various sequences are found along columns and diagonals:
T(n,0) = A355357(n) = [1, 1, 1, 4, 7, 20, 43, 110, 262, 674, 1684, ...],
T(n+2,1) = A359725(n) = [2, 5, 21, 51, 170, 454, 1367, 3776, 11062, ...],
T(n+3,2) = A359726(n) = [1, 9, 49, 179, 711, 2390, 8361, 27082, 89389, ...],
T(3*n,2*n) = A000108(n) = [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, ...],
T(3*n+1,2*n) = A359722(n) = [1, 9, 54, 269, 1254, 5642, 24828, 107613, ...],
T(3*n+2,2*n+1) = A097613(n+2) = [2, 7, 25, 91, 336, 1254, 4719, 17875, ...].

Links

Crossrefs

Cf. A359721 (row sums), A357797 (y=2), A359723 (y=3), A359724 (y=4).
Cf. A355357 (T(n,0)), A359725 (T(n+2,1)), A359726 (T(n+3,2)).
Cf. A000108 (T(3*n,2*n)), A097613 (T(3*n+2,2*n+1)), A359722 (T(3*n+1,2*n)).

Programs

  • PARI
    /* Print this irregular triangle */
{T(n,k) = 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 * (y + x^n +x*O(x^#A) )^n * Ser(A)^n ), #A-1) );
polcoeff(A[n+1],k,y)}
for(n=0, 15, for(k=0, (2*n)\3, print1(T(n,k), ", "));print(""))

Formula

G.f.: A(x,y) = Sum_{n>=0} Sum_{k=0..floor(2*n/3)} T(n,k)*x^n*y^k may be described by the following.
(1) x = Sum_{n=-oo..+oo} (-1)^n * x^n * (y + x^n)^n * A(x,y)^n.
(2) x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / ( (1 + y*x^n)^n * A(x,y)^n ).
T(3*n,2*n) = binomial(2*n+1,n)/(2*n+1) = A000108(n), n >= 0 (Catalan numbers).
T(3*n+2,2*n+1) = binomial(2*n+1,n+1) + binomial(2*n+2,n) = A097613(n+2), for n >= 0.

A357797 a(n) = coefficient of x^n in the power series A(x) such that: x = Sum_{n=-oo..+oo} (-1)^n * x^n * (2 + x^n)^n * A(x)^n.

Original entry on oeis.org

1, 1, 5, 18, 85, 374, 1659, 7774, 36876, 177494, 867424, 4285653, 21373782, 107475746, 544244911, 2773091748, 14207171278, 73140904609, 378184133959, 1963127909395, 10226682384980, 53446907352828, 280150058149086, 1472424136948438, 7758105323877698, 40970959715619200, 216830651728330127
Offset: 0

Views

Author

Paul D. Hanna, Dec 22 2022

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 + 5*x^2 + 18*x^3 + 85*x^4 + 374*x^5 + 1659*x^6 + 7774*x^7 + 36876*x^8 + 177494*x^9 + 867424*x^10 + 4285653*x^11 + 21373782*x^12 + ...

Links

Crossrefs

Cf. A359720, A359721, A359723, A357798.

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

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.

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

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 + 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.
Showing 1-4 of 4 results.

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