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 1-3 of 3 results.

A299427 Square table where T(n,k) = binomial(n*(n+k), k) * n/(n+k), for n>=1, k>=0, as read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 4, 1, 1, 9, 14, 1, 1, 16, 63, 48, 1, 1, 25, 184, 408, 165, 1, 1, 36, 425, 1872, 2565, 572, 1, 1, 49, 846, 6175, 17980, 15939, 2002, 1, 1, 64, 1519, 16536, 82775, 167552, 98670, 7072, 1, 1, 81, 2528, 38318, 292581, 1059380, 1535352, 610740, 25194, 1, 1, 100, 3969, 79808, 861175, 4874688, 13177125, 13934752, 3786588, 90440, 1
Offset: 0

Views

Author

Paul D. Hanna, Feb 19 2018

Keywords

Examples

			This table begins:
n=1: [1,  1,    1,      1,       1,         1,          1, ...];
n=2: [1,  4,   14,     48,     165,       572,       2002, ...];
n=3: [1,  9,   63,    408,    2565,     15939,      98670, ...];
n=4: [1, 16,  184,   1872,   17980,    167552,    1535352, ...];
n=5: [1, 25,  425,   6175,   82775,   1059380,   13177125, ...];
n=6: [1, 36,  846,  16536,  292581,   4874688,   78119454, ...];
n=7: [1, 49, 1519,  38318,  861175,  18008676,  358919022, ...];
n=8: [1, 64, 2528,  79808, 2214640,  56592320, 1367090208, ...];
n=9: [1, 81, 3969, 153117, 5132565, 157000275, 4507103601, ...];
...
Row generating functions R(x,n)^(n^2) begin:
R(x,1) = 1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + ...
R(x,2)^4 = 1 + 4*x + 14*x^2 + 48*x^3 + 165*x^4 + 572*x^5  + ...
R(x,3)^9 = 1 + 9*x + 63*x^2 + 408*x^3 + 2565*x^4 + 15939*x^5 + ...
R(x,4)^16 = 1 + 16*x + 184*x^2 + 1872*x^3 + 17980*x^4 + 167552*x^5 + ...
R(x,5)^25 = 1 + 25*x + 425*x^2 + 6175*x^3 + 82775*x^4 + 1059380*x^5 + ...
R(x,6)^36 = 1 + 36*x + 846*x^2 + 16536*x^3 + 292581*x^4 + 4874688*x^5 + ...
...
Related series R(x,n) = 1 + x*R(x,n)^n begin:
R(x,1) = 1 + x + x^2 + x^3 + x^4 + x^5 + ...
R(x,2) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + ...
R(x,3) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + ...
R(x,4) = 1 + x + 4*x^2 + 22*x^3 + 140*x^4 + 969*x^5 + ...
R(x,5) = 1 + x + 5*x^2 + 35*x^3 + 285*x^4 + 2530*x^5 + ...
R(x,6) = 1 + x + 6*x^2 + 51*x^3 + 506*x^4 + 5481*x^5 + ...
...
where R(x,n)^m = Sum_{k>=0} C(m + n*k, k) * m/(m + n*k) * x^k.
...

Links

Crossrefs

Cf. A299044 (antidiagonal sums), A299428 (diagonal), A299429.

Programs

  • PARI
    {T(n,k) = binomial(n*(n+k), k) * n/(n+k) }
/* Print as a square table of first 9 rows */
for(n=1,9,print1("n="n": [",); for(k=0,8,print1(T(n,k),", ")); print1("...];");print(""))
/* Print as a Flattened table read by antidiagonals */
for(n=1,10,for(k=0,n,print1(T(n-k+1,k),", ")))

Formula

G.f. for row n: R(x,n)^(n^2) = Sum_{k>=0} C(n*(n+k), k) * n/(n+k) * x^k, where R(x,n) = 1 + x*R(x,n)^n.

A299428 a(n) = binomial((n+1)*(2*n+1), n) * (n+1)/(2*n+1).

Original entry on oeis.org

1, 4, 63, 1872, 82775, 4874688, 358919022, 31726703424, 3273365223135, 386120802767700, 51255818495200660, 7561964058268969440, 1227474574989496660008, 217398508335873934190800, 41718377034325560258265500, 8622580886584109407750765824, 1909661474657747399115123743055, 451173386162679212279972033149500
Offset: 0

Views

Author

Paul D. Hanna, Feb 19 2018

Keywords

Comments

Main diagonal of square table A299427.

Links

Crossrefs

Cf. A299427, A299429.

Programs

  • PARI
    {a(n) = binomial((n+1)*(2*n+1), n) * (n+1)/(2*n+1)}
for(n=0,20,print1(a(n),", "))

Formula

a(n) ~ 2^(n - 3/2) * exp(n + 5/4) * n^(n - 1/2) / sqrt(Pi). - Vaclav Kotesovec, Feb 19 2018

Extensions

a(15) corrected by Seiichi Manyama, Feb 10 2019

A352700 G.f.: Sum_{n>=0} binomial((n+1)*(2*n+1),n)/(2*n+1) * x^n / C(x)^(n*(2*n+1)+1), where C(x) = 1 + x*C(x)^2 is the Catalan function (A000108).

Original entry on oeis.org

1, 1, 12, 239, 7178, 296092, 15666162, 1014796995, 77899495174, 6919858148750, 698584345392968, 79022119891573410, 9902447587480555624, 1361894352334815968554, 203969111022547680433454, 33047362680815865252524643, 5759708920548423261284008230
Offset: 0

Views

Author

Paul D. Hanna, Mar 29 2022

Keywords

Comments

Compare g.f. to: 1 = Sum_{n>=0} binomial((k+1)*(2*n+1),n)/(2*n+1) * x^n / C(x)^(k*(2*n+1)+1) which holds for fixed k, where C(x) = 1 + x*C(x)^2 is the Catalan function (A000108).
The g.f. A(x) seems to satisfy A(x)^3 = A(x^3) (mod 3); compare this to the congruence: C(x)^3 = C(x^3) (mod 3), where C(x) is the Catalan function.
Odd terms seem to occur only at positions 2^n-1 for n >= 0.
Conjectures: given g.f. A(x), let C(x) = (1 - sqrt(1-4*x))/(2*x) be the Catalan power series (A000108), then
(1) A(x)^3 = A(x^3) (mod 3),
(2) A(x) = C(x) + x^2*C(x)^3 (mod 3) = (2 - x)*C(x) - 1 (mod 3),
(3) A(x) = C(x) (mod 2),
(4) a(n) = binomial(2*n+1,n)/(2*n+1) + 3*binomial(2*n-1,n-2)/(2*n-1) (mod 3) for n >= 0,
(5) a(n) = 2*A000108(n) - A000108(n-1) (mod 3) for n >= 1,
(6) a(n) = A000108(n) (mod 2) for n >= 0.

Examples

			G.f.: A(x) = 1 + x + 12*x^2 + 239*x^3 + 7178*x^4 + 296092*x^5 + 15666162*x^6 + 1014796995*x^7 + 77899495174*x^8 + 6919858148750*x^9 + ...
where
A(x) = 1/C(x) + 2*x/C(x)^4 + 21*x^2/C(x)^11 + 468*x^3/C(x)^22 + 16555*x^4/C(x)^37 + 812448*x^5/C(x)^56 + 51274146*x^6/C(x)^79 + 3965837928*x^7/C(x)^106 + ... + (n+1)*A299429(n)*x^n/C(x)^(n*(2*n+1)+1) + ...
and
C(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 + 429*x^7 + 1430*x^8 + 4862*x^9 + ... + A000108(n)*x^n + ...
Congruence modulo 3.
(1) It appears that A(x)^3 is congruent to A(x^3) modulo 3, where
A(x)^3 = 1 + 3*x + 39*x^2 + 790*x^3 + 23436*x^4 + 949701*x^5 + 49503687*x^6 + 3171679536*x^7 + 241578165750*x^8 + 21340270771814*x^9 + ...
and
(A(x)^3 - A(x^3))/3 = x + 13*x^2 + 263*x^3 + 7812*x^4 + 316567*x^5 + 16501225*x^6 + 1057226512*x^7 + 80526055250*x^8 + 7113423590525*x^9 + ...
(2) Also, g.f. A(x) seems to be congruent to C(x) + x^2*C(x)^3, where
C(x) + x^2*C(x)^3 = 1 + x + 3*x^2 + 8*x^3 + 23*x^4 + 70*x^5 + 222*x^6 + 726*x^7 + 2431*x^8 + 8294*x^9 + ... + (C(2*n,n)/(n+1) + C(2*n-1,n-2)*3/(2*n-1))*x^n + ...

Crossrefs

Cf. A298696, A299429, A000108, A000782.

Programs

  • PARI
    {a(n) = my(C = (1 - sqrt(1-4*x +O(x^(n+3))))/(2*x),
A = sum(m=0,n, binomial((m+1)*(2*m+1),m)/(2*m+1) * x^m/C^(m*(2*m+1)+1))); polcoeff(A,n)}
for(n=0,20,print1(a(n),", "))

Formula

G.f. A(x) satisfies:
(1) A(x-x^2) = Sum_{n>=0} binomial((n+1)*(2*n+1),n)/(2*n+1) * (x/(1-x))^n * (1-x)^((n+1)*(2*n+1)).
(2) A(x/(1+x)^2) = Sum_{n>=0} binomial((n+1)*(2*n+1),n)/(2*n+1) * x^n / (1+x)^((n+1)*(2*n+1)).
Showing 1-3 of 3 results.

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