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.

A047098 a(n) = 2*binomial(3*n, n) - Sum_{k=0..n} binomial(3*n, k).

Original entry on oeis.org

1, 2, 8, 38, 196, 1062, 5948, 34120, 199316, 1181126, 7080928, 42860534, 261542752, 1607076200, 9934255472, 61732449648, 385393229460, 2415935640198, 15200964233864, 95962904716402, 607640599286276, 3858198001960438, 24559243585545644, 156692889782067712
Offset: 0

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Author

Clark Kimberling, Aug 15 1998

Keywords

Comments

T(2n,n), array T as in A047089. [Corrected Dec 08 2006]
Let B_3^+ denote the semigroup with presentation . Let D=aba be the 'fundamental word'. Then this sequence is also equal to the number of words in B_3^+ equal in B_3^+ to D^n, n >= 0. - Stephen P. Humphries, Jan 20 2004
In the language of Riordan arrays, row sums of (1/(1+x), x/(1+x)^3)^-1, where (1/(1+x), x/(1+x)^3) has general term (-1)^(n-k)*binomial(n+2k, 3k). - Paul Barry, May 09 2005
Hankel transform is 2^n*A051255(n) where A051255 is the Hankel transform of C(3n,n)/(2n+1). - Paul Barry, Jan 21 2007

Links

Crossrefs

Column k=2 of A213028.
Cf. A047089, A047099, A107026, A321957.

Programs

  • Maple
    A047098 := n -> 2*binomial(3*n, n)-add(binomial(3*n, k), k=0..n);
  • Mathematica
    Table[2Binomial[3n,n]-Sum[Binomial[3n,k],{k,0,n}],{n,0,35}] (* Harvey P. Dale, Jul 27 2011 *)
  • PARI
    a(n)=if(n<0,0,polcoeff((((1+10*x-2*x^2)+(1-4*x)*sqrt(1-4*x+x*O(x^n)))/2)^n,n))
  • PARI
    a(n)=if(n<0,0, 2*binomial(3*n,n)-sum(k=0,n,binomial(3*n,k)))

Formula

G.f. A(x)=y satisfies (8x-1)y^3-y^2+y+1=0. - Michael Somos, Jan 28 2004
Coefficient of x^n in ((1+10x-2x^2+(1-4x)^(3/2))/2)^n. - Michael Somos, Sep 25 2003
a(n) = Sum_{k = 0..n} A109971(k)*2^k; a(0) = 1, a(n) = Sum_{k = 0..n} 2^k*C(3n-k,n-k)*2*k/(3*n-k), n > 0. - Paul Barry, Jan 21 2007
Conjecture: 2*n*(2*n-1)*a(n) +(-71*n^2+112*n-48)*a(n-1) +3*(131*n^2-391*n+296)*a(n-2) -72*(3*n-7)*(3*n-8)*a(n-3)=0. - R. J. Mathar, Nov 30 2012
a(n) = A321957(n) + 2*binomial(3*n, n) - 8^n. - Peter Luschny, Nov 22 2018
The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and positive integers n and k. - Peter Bala, Mar 05 2022

Extensions

Clark Kimberling, Dec 08 2006, changed "T(3n,2n)" to "T(2n,n)" in the comment line, but observes that some of the other comments seem to apply to the sequence T(3n,2n) rather than to the sequence T(2n,n).
Edited by N. J. A. Sloane, Dec 21 2006, replacing the old definition in terms of A047089 by an explicit formula supplied by Benoit Cloitre, Oct 25 2003.

A307468 Cogrowth sequence for the Heisenberg group.

Original entry on oeis.org

1, 4, 28, 232, 2156, 21944, 240280, 2787320, 33820044, 424925872, 5486681368, 72398776344, 972270849512, 13247921422480, 182729003683352, 2546778437385032, 35816909974343308, 507700854900783784, 7246857513425470288, 104083322583897779656
Offset: 0

Views

Author

Igor Pak, Apr 09 2019

Keywords

Comments

This is the number of words of length 2n in the letters x,x^{-1},y,y^{-1} that equal the identity of the Heisenberg group H=.
Also, this is the number of closed walks of length 2n on the square lattice enclosing algebraic area 0 [Béguin et al.]. - Andrey Zabolotskiy, Sep 15 2021

Examples

			For n=1 the a(1)=4 words are x^{-1}x, xx^{-1}, y^{-1}y, yy^{-1}.

Links

Crossrefs

Related cogrowth sequences: Z A000984, Z^2 A002894, Z^3 A002896, (Z/kZ)^*2 for k = 2..5: A126869, A047098, A107026, A304979, Richard Thompson's group F A246877. The cogrowth sequences for BS(N,N) for N = 2..10 are A229644, A229645, A229646, A229647, A229648, A229649, A229650, A229651, A229652.
Cf. A352838, A178106.

Formula

Asymptotics: a(n) ~ (1/2) * 16^n * n^(-2).

A304979 The nonzero terms of the cogrowth sequence of (Z/5Z)^*2 = * with respect to the generating set {(x,1), (1,y)}.

Original entry on oeis.org

1, 2, 12, 92, 792, 7302, 70464, 702536, 7178568, 74771570, 790906012, 8472417384, 91724327928, 1001987961834, 11030476949952, 122247789508992, 1362840516623944, 15272530735735338, 171946029518128956, 1943927810200670820, 22059590401383177792, 251183781609841838444
Offset: 0

Views

Author

Marni Mishna, May 22 2018

Keywords

Comments

The number of paths on the Cayley graph of ((Z/5Z)^*2, {(x,1), (1,y)} which start and end at the identity.
a(n) is the number of words of length 5n over the alphabet {x,y} that reduce to the empty string upon iteratively removing factors of x^5 and y^5. For n=2, the a(2)=12 words of length 10 that reduce to the empty string are x^10, y^10, x^i y^5 x^(5-i), for i=1..5, and y^i x^5 y^(5-i), for i=1..5.

Links

Crossrefs

Related cogrowth sequences (Z/2Z)^*2: A126869; (Z/3Z)^*2: A047098; (Z/4Z)^*2: A107026.

Programs

  • Mathematica
    terms = 22;
A[] = 0; Do[A[x] = (1 + 4 A[x] + 6 A[x]^2 + 4 A[x]^3 + A[x]^4 + 32 x A[x]^5)/(1 + A[x])^4 + O[x]^terms // Normal, {terms}];
CoefficientList[A[x], x] (* Jean-François Alcover, Nov 16 2018 *)

Formula

G.f.: A(x) satisfies 32*x*A(x)^5 - (A(x)-1)*(A(x)+1)^4 = 0.
a(n) satisfies the recurrence (2120000*(5*n+1))*(5*n+2)*(5*n+3)*(5*n+4)*a(n) + (250*(160980199*n^4 + 1129209134*n^3 + 2872721885*n^2 + 3155706646*n + 1267579560))*a(n+1) - (50*(109722203*n^4 + 959367613*n^3 + 3144281425*n^2 + 4572924587*n + 2485585548))*a(n+2) + (60*(4290021*n^4 + 51502996*n^3 + 243316306*n^2 + 532456081*n + 451079946))*a(n+3) - (3*(2673299*n^4 + 44756419*n^3 + 283571239*n^2 + 805783469*n + 866093430))*a(n+4) + (4008*(n+5))*(4*n+17)*(2*n+9)*(4*n+19)*a(n+5) = 0.
a(n) ~ 5^(5*n + 1/2) / (9 * sqrt(Pi) * n^(3/2) * 2^(8*n - 3/2)). - Vaclav Kotesovec, Oct 24 2023
a(n) = binomial(5*n,n) - 3 * Sum_{k=0..n-1} binomial(5*n,k). - Seiichi Manyama, Apr 05 2024

A382100 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of 1/(2 - B_k(x)), where B_k(x) = 1 + x*B_k(x)^k.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 4, 1, 1, 1, 4, 10, 8, 1, 1, 1, 5, 19, 35, 16, 1, 1, 1, 6, 31, 98, 126, 32, 1, 1, 1, 7, 46, 213, 531, 462, 64, 1, 1, 1, 8, 64, 396, 1556, 2974, 1716, 128, 1, 1, 1, 9, 85, 663, 3651, 11843, 17060, 6435, 256, 1
Offset: 0

Views

Author

Seiichi Manyama, Mar 15 2025

Keywords

Examples

			Square array begins:
  1,  1,   1,    1,     1,     1,     1, ...
  1,  1,   1,    1,     1,     1,     1, ...
  1,  2,   3,    4,     5,     6,     7, ...
  1,  4,  10,   19,    31,    46,    64, ...
  1,  8,  35,   98,   213,   396,   663, ...
  1, 16, 126,  531,  1556,  3651,  7391, ...
  1, 32, 462, 2974, 11843, 35232, 86488, ...

Crossrefs

Columns k=0..5 give A000012, A011782, A088218, A047099 (for n > 0), A107026(n)/2 (for n > 0), A304979(n)/2 (for n > 0).
Cf. A107027, A107030, A355262, A382101.

Programs

  • PARI
    a(n, k) = polcoef(1/(2-sum(j=0, n, binomial(k*j+1, j)/(k*j+1)*x^j+x*O(x^n))), n);
  • PARI
    a(n, k) = if(n==0, 1, (binomial(k*n, n)-(k-2)*sum(j=0, n-1, binomial(k*n, j)))/2);

Formula

A(n,k) = ( binomial(k*n,n) - (k-2) * Sum_{j=0..n-1} binomial(k*n,j) )/2 for n > 0.
G.f. of column k: 1/( 1 - Series_Reversion( x/(1+x)^k ) ).
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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