A006605 -id:A006605 - cp's OEIS frontend

A143927 G.f. satisfies: A(x) = (1 + xA(x) + x^2A(x)^2)^2.

1, 2, 7, 28, 123, 572, 2769, 13806, 70414, 365636, 1926505, 10273870, 55349155, 300783420, 1646828655, 9075674700, 50304255210, 280248358964, 1568399676946, 8813424968192, 49709017472751, 281306750922072, 1596802663432503

Offset: 0

Paul D. Hanna, Sep 08 2008

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000
Vladimir Kruchinin and D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.

Cf. A006605, A027908, A143926.

Table[GegenbauerC[n,-2n-2,-1/2]/(n+1),{n,0,12}] (* Emanuele Munarini, Oct 20 2016 *)
n = 20;
A = Sum[a[k] x^k, {k, 0, n}] + x O[x]^n;
Table[a[k], {k, 0, n}] /. Reverse[Solve[LogicalExpand[(1 + x A + x^2 A^2)^2 == A]]] (* Emanuele Munarini, Oct 20 2016 *)

makelist(ultraspherical(n,-2*n-2,-1/2)/(n+1),n,0,12); /* Emanuele Munarini, Oct 20 2016 */

{a(n)=local(A=1+x+x*O(x^n));for(i=0,n,A=(1+x*A+x^2*A^2)^2);polcoeff(A,n)}

Self-convolution of A006605.
Bisection of A143926.
a(n) = ((24*n+12)*A006605(n) + (3*n+5)*A006605(n+1))/(13*n+17). - Mark van Hoeij, Jul 14 2010
a(n) = (1/(n+1))*Sum_{j=0..2*n+2} (binomial(j,2*j-3*n-4)*binomial(2*n+2 ,j)). - Vladimir Kruchinin, Dec 24 2010
a(n) = GegenbauerPoly(n,-2n-2,-1/2)/(n+1). - Emanuele Munarini, Oct 20 2016
a(n) = T(2*n+2, n)/(n+1), where T(n,k) = A027907(n,k). - Emanuele Munarini, Oct 20 2016
The g.f. A(x) satisfies 1 + x*A'(x)/A(x) = 1 + 2*x + 10*x^2 + 50*x^3 + 266*x^3 + ..., the g.f. of A027908. - Peter Bala, Aug 03 2023

A365183 G.f. satisfies A(x) = 1 + xA(x)^4(1 + x*A(x)^4).

Original entry on oeis.org

1, 1, 5, 34, 268, 2299, 20838, 196326, 1903524, 18868861, 190356231, 1948055058, 20173907384, 211020478270, 2226243632838, 23660868061422, 253099278807684, 2722819049879436, 29439894433161189, 319749417998303470, 3486914150183526920

Offset: 0

Seiichi Manyama, Aug 25 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..939
Jun Yan, Lattice paths enumerations weighted by ascent lengths, arXiv:2501.01152 [math.CO], 2025. See p. 13.

Cf. A002294, A365178, A365180, A365181, A365182.
Cf. A006605, A255673, A365189.
Cf. A364989.

a(n) = sum(k=0, n\2, binomial(n-k, k)*binomial(4*n+1, n-k))/(4*n+1);

a(n) = (1/(4*n+1)) * Sum_{k=0..floor(n/2)} binomial(n-k,k) * binomial(4*n+1,n-k).

A255673 Coefficients of A(x), which satisfies: A(x) = 1 + xA(x)^3 + x^2A(x)^6.

Original entry on oeis.org

1, 1, 4, 21, 127, 833, 5763, 41401, 305877, 2309385, 17739561, 138197876, 1089276972, 8670856834, 69606939717, 562879492551, 4580890678781, 37490975387565, 308369889858450, 2547741413147700, 21133987935358776, 175947462569886786, 1469656053534121804

Offset: 0

Werner Schulte, Jul 10 2015

nonn

This sequence is the next after A001006 and A006605.

			A(x) = 1 + x + 4*x^2 + 21*x^3 + 127*x^4 + 833*x^5 + 5763*x^6 ...

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Jun Yan, Lattice paths enumerations weighted by ascent lengths, arXiv:2501.01152 [math.CO], 2025. See p. 13.

Cf. A001006, A006605.

Cf. A365183, A365189.

a:= n-> coeff(series(RootOf(1-A+x*A^3+x^2*A^6, A), x, n+1), x, n):
seq(a(n), n=0..30);  # Alois P. Heinz, Jul 15 2015
# second Maple program:
a:= proc(n) option remember; `if`(n<2, 1, 9*(((3*n-1))*
     (2*n-1)*(3*n-2)*(9063*n^4-18126*n^3+8403*n^2+660*n-280)*a(n-1)
     +(27*(n-1))*(3*n-1)*(3*n-4)*(3*n-2)*(3*n-5)*(57*n^2-2)*a(n-2))
      /((5*(5*n+2))*(5*n-1)*(5*n+1)*(5*n-2)*n*(57*n^2-114*n+55)))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Jul 16 2015

m = 30; A[_] = 0;
Do[A[x_] = 1 + x A[x]^3 + x^2 A[x]^6 + O[x]^m, {m}];
CoefficientList[A[x], x] (* Jean-François Alcover, Oct 04 2019 *)

a(n) = sum(k=0, n\2, binomial(n-k, k)*binomial(3*n+1, n-k))/(3*n+1); \\ Seiichi Manyama, Sep 02 2023

a(n+1) = Sum_{j=0..3*n+4} binomial(j,2*j-5*n-7) * binomial(3*n+4,j) / (3*n+4). (conjectured). [Vladimir Kruchinin, Mar 09 2013]
a(n) = 1/(3*n+1) * Sum_{k=0..n} (-1)^k * binomial(3*n+1, k) * binomial(6*n+2-2*k, n-k). (conjectured)
G.f. A(x) satisfies A(x) = G(x*A(x)), where G is g.f. of A006605.
G.f. A(x) satisfies A(x) = H(x*A(x)^2), where H is g.f. of A001006.
From Peter Bala, Jul 27 2023: (Start)
Define b(n) = [x^n] (1 + x + x^2)^(3*n). Then A(x)^3 = exp(Sum_{n >= 1} b(n)*x^n/n).
A(x^3) = (1/x) * series reversion of x/(1 + x^3 + x^6) = 1 + x^3 + 4*x^6 + 21*x^9 + 127*x^12 + .... (End)
a(n) = (1/(3*n+1)) * Sum_{k=0..floor(n/2)} binomial(n-k,k) * binomial(3*n+1,n-k). - Seiichi Manyama, Sep 02 2023

More terms from Alois P. Heinz, Jul 15 2015

A365189 G.f. satisfies A(x) = 1 + xA(x)^5(1 + x*A(x)^5).

Original entry on oeis.org

1, 1, 6, 50, 485, 5130, 57391, 667777, 7999095, 97986680, 1221813880, 15456556791, 197887386913, 2559189842240, 33383097891135, 438714241508615, 5803049210371375, 77199163872173757, 1032215519193531310, 13864180990526161995, 186975433988014039830

Offset: 0

Seiichi Manyama, Aug 25 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..864
Jun Yan, Lattice paths enumerations weighted by ascent lengths, arXiv:2501.01152 [math.CO], 2025. See p. 13.

Cf. A002295, A365184, A365185, A365186, A365187, A365188.

Cf. A006605, A255673, A365183.

a(n) = sum(k=0, n\2, binomial(n-k, k)*binomial(5*n+1, n-k))/(5*n+1);

a(n) = (1/(5*n+1)) * Sum_{k=0..floor(n/2)} binomial(n-k,k) * binomial(5*n+1,n-k).

A143926 G.f. satisfies: A(x) = 1 + xA(x)A(-x) + x^2A(x)^2A(-x)^2.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 7, 11, 28, 46, 123, 207, 572, 979, 2769, 4797, 13806, 24138, 70414, 123998, 365636, 647615, 1926505, 3428493, 10273870, 18356714, 55349155, 99229015, 300783420, 540807165, 1646828655, 2968468275, 9075674700

Offset: 0

Paul D. Hanna, Sep 06 2008

nonn

Bisections form A006605 and A143927;
A006605 is the number of modes of connections of 2n points and
A143927 is the self-convolution of A006605.

			G.f. A(x) = 1 + x + x^2 + x^3 + 2*x^4 + 3*x^5 + 7*x^6 + 11*x^7 +...
A(x)*A(-x) = 1 + x^2 + 3*x^4 + 11*x^6 + 46*x^8 + 207*x^10 + 979*x^12 +...
A(x)^2*A(-x)^2 = 1 + 2*x^2 + 7*x^4 + 28*x^6 + 123*x^8 + 572*x^10 +...
A(x)^4*A(-x)^4 = 1 + 4*x^2 + 18*x^4 + 84*x^6 + 407*x^8 + 2028*x^10 +...
from this we see that if B(x^2) = A(x)*A(-x)
then B(x) = 1 + x*B(x)^2 + x^2*B(x)^4
and A(x) = 1 + x*B(x^2) + x^2*B(x^2)^2.

Vaclav Kotesovec, Table of n, a(n) for n = 0..100

Cf. A006605, A143927.

a[n_] := Module[{A = 1 + x, B}, For[i = 0, i <= n, i++, B = A*(A /. x -> -x); A = 1 + x*B + x^2*B^2 + O[x]^(n+1) // Normal]; SeriesCoefficient[A, {x, 0, n}]]; Table[a[n], {n, 0, 32}] (* Jean-François Alcover, Oct 22 2016, adapted from PARI *)

{a(n)=local(A=1+x+x*O(x^n));for(i=0,n,B=A*subst(A,x,-x);A=1+x*B+x^2*B^2);polcoeff(A,n)}

Define B(x) by B(x^2) = A(x)*A(-x); then B(x) = 1 + x*B(x)^2 + x^2*B(x)^4 is the g.f. of A006605.
Recurrence: 3*n*(3*n+1)*(3*n+2)*(507*n^4 - 3575*n^3 + 8895*n^2 - 8953*n + 3054)*a(n) =  - 12*(4017*n^5 - 20319*n^4 + 31895*n^3 - 17595*n^2 + 2338*n + 384)*a(n-1) + 4*(n-2)*(17745*n^6 - 125125*n^5 + 331891*n^4 - 396335*n^3 + 173912*n^2 + 17532*n - 13140)*a(n-2) - 144*(n-3)*(n-2)*(312*n^3 - 988*n^2 + 407*n + 29)*a(n-3) + 144*(n-4)*(n-3)*(n-2)*(507*n^4 - 1547*n^3 + 1212*n^2 + 140*n - 72)*a(n-4). - Vaclav Kotesovec, Dec 21 2013
a(n) ~ c*d^n/(sqrt(Pi)*n^(3/2)), where d = sqrt(70/27+(26*sqrt(13))/27) = 2.4626418602647616787... is the root of the equation -144 - 140*d^2 + 27*d^4 = 0 and c = 2*sqrt((5+1/sqrt(13))/3)/3 = 0.88421131194123... if n is even, and c = sqrt(1+11/sqrt(13))/3 = 0.670890873659690... if n is odd. - Vaclav Kotesovec, Dec 21 2013

A026302 a(n) = number of (s(0), s(1), ..., s(n)) such that s(i) is a nonnegative integer and |s(i) - s(i-1)| <= 1 for i = 1,2,...,n, s(0) = 0, s(2n) = n. Also a(n) = T(2n,n), where T is the array in A026300.

Original entry on oeis.org

1, 2, 9, 44, 230, 1242, 6853, 38376, 217242, 1239980, 7123765, 41141916, 238637282, 1389206210, 8112107475, 47495492400, 278722764954, 1638970147188, 9654874654438, 56965811111240, 336590781348276, 1991357644501170

Offset: 0

Clark Kimberling

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1280
D. Kruchinin and V. Kruchinin, A Method for Obtaining Generating Function for Central Coefficients of Triangles, Journal of Integer Sequence, Vol. 15 (2012), article 12.9.3.

Bisection of A026307.

b:= proc(x, y) option remember; `if`(min(x, y)<0, 0,
     `if`(max(x, y)=0, 1, b(x-1, y)+b(x, y-1)+b(x-2, y+1)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..23);  # Alois P. Heinz, Sep 28 2019

Table[Binomial[2*n, n]*Hypergeometric2F1[1/2 - n/2, -n/2, 2 + n, 4], {n, 0, 30}] (* Vaclav Kotesovec, Sep 17 2019 *)

A026300(n,k)={ if(n<0 || k < 0, return(0) ;) ; if(n<=1, 1, if(k==0, 1, sum(i=0,k/2, binomial(n,2*i+n-k)*(binomial(2*i+n-k,i)-binomial(2*i+n-k,i-1))) ;) ;) ; }
A026302(n)={ A026300(2*n,n) ; }
{ for(n=0,21, print(n," ",A026302(n))) ; } \\ R. J. Mathar, Oct 26 2006

a(n) = binomial(2*n,n)*hypergeom([ -n/2, 1/2 - n/2],[n+2],4). - Mark van Hoeij, Jun 02 2010
a(n) = (n + 1) * A006605(n). - Mark van Hoeij, Jul 02 2010
G.f. A(x)=(x*M(x))', where M(x)=1+x*M(x)^2+x^2*M(x)^4. - Vladimir Kruchinin, May 25 2012
From Ilya Gutkovskiy, Sep 21 2017: (Start)
a(n) = [x^n] ((1 - x - sqrt(1 - 2*x - 3*x^2))/(2*x^2))^(n+1).
a(n) = [x^n] (1/(1 - x - x^2/(1 - x - x^2/(1 - x - x^2/(1 - x - x^2/(1 - x - x^2/(1 - ...)))))))^(n+1), a continued fraction. (End)
From Vaclav Kotesovec, Sep 17 2019: (Start)
Recurrence: 3*n^2*(3*n + 1)*(3*n + 2)*(13*n - 9)*a(n) = 2*(n+1)*(2*n - 1)*(455*n^3 - 315*n^2 - 44*n + 24)*a(n-1) + 36*n*(n+1)*(2*n - 3)*(2*n - 1)*(13*n + 4)*a(n-2).
a(n) ~ sqrt(277 + 89*sqrt(13)) * (70 + 26*sqrt(13))^n / (13^(1/4) * sqrt(2*Pi*n) * 3^(3*n + 5/2)). (End)

Corrected by R. J. Mathar, Oct 26 2006

A365268 G.f. satisfies A(x) = 1 + xA(x)^2(1 + x^3*A(x)^2).

Original entry on oeis.org

1, 1, 2, 5, 15, 48, 160, 549, 1929, 6909, 25134, 92612, 344924, 1296376, 4910656, 18728645, 71857133, 277160183, 1074085446, 4180057725, 16329796959, 64014638564, 251734985808, 992788252700, 3925688845948, 15560762343388, 61818928594952

Offset: 0

Seiichi Manyama, Aug 30 2023

nonn

Cf. A006605, A049140.

Cf. A063021, A365267.

a(n) = sum(k=0, n\4, binomial(n-3*k, k)*binomial(2*n-4*k+1, n-3*k)/(2*n-4*k+1));

a(n) = Sum_{k=0..floor(n/4)} binomial(n-3*k,k) * binomial(2*n-4*k+1,n-3*k)/(2*n-4*k+1).

A378292 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,0) = 0^n and T(n,k) = k * Sum_{r=0..n} binomial(2n+k,r) binomial(r,n-r)/(2*n+k) for k > 0.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 3, 7, 11, 0, 1, 4, 12, 28, 46, 0, 1, 5, 18, 52, 123, 207, 0, 1, 6, 25, 84, 240, 572, 979, 0, 1, 7, 33, 125, 407, 1155, 2769, 4797, 0, 1, 8, 42, 176, 635, 2028, 5733, 13806, 24138, 0, 1, 9, 52, 238, 936, 3276, 10332, 29136, 70414, 123998, 0

Offset: 0

Seiichi Manyama, Nov 21 2024

			Square array begins:
  1,   1,    1,    1,     1,     1,     1, ...
  0,   1,    2,    3,     4,     5,     6, ...
  0,   3,    7,   12,    18,    25,    33, ...
  0,  11,   28,   52,    84,   125,   176, ...
  0,  46,  123,  240,   407,   635,   936, ...
  0, 207,  572, 1155,  2028,  3276,  4998, ...
  0, 979, 2769, 5733, 10332, 17140, 26860, ...

Columns k=0..2 give A000007, A006605, A143927.

Cf. A009766, A026300, A378289, A378290, A378291.

T(n, k, t=2, u=2) = if(k==0, 0^n, k*sum(r=0, n, binomial(t*r+u*(n-r)+k, r)*binomial(r, n-r)/(t*r+u*(n-r)+k)));
matrix(7, 7, n, k, T(n-1, k-1))

G.f. A_k(x) of column k satisfies A_k(x) = ( 1 + x * A_k(x)^(2/k) * (1 + x * A_k(x)^(2/k)) )^k for k > 0.
G.f. of column k: B(x)^k where B(x) is the g.f. of A006605.
B(x)^k = B(x)^(k-1) + x * B(x)^(k+1) + x^2 * B(x)^(k+3). So T(n,k) = T(n,k-1) + T(n-1,k+1) + T(n-2,k+3) for n > 1.

A364477 G.f. satisfies A(x) = 1 + xA(x)^2 + x^2A(x)^7.

Original entry on oeis.org

1, 1, 3, 14, 76, 448, 2791, 18078, 120516, 821435, 5698422, 40101623, 285583775, 2054272430, 14903954415, 108932920861, 801350333186, 5928653489398, 44084056075057, 329279673851792, 2469493161891742, 18588339309502760, 140383789476473354

Offset: 0

Seiichi Manyama, Jul 26 2023

nonn

Cf. A001002, A001764, A006605, A052709, A143330, A364473.

Cf. A364476.

a(n) = sum(k=0, n\2, binomial(2*n+3*k, k)*binomial(2*n+2*k, n-2*k)/(n+4*k+1));

a(n) = Sum_{k=0..floor(n/2)} binomial(2*n+3*k,k) * binomial(2*n+2*k,n-2*k) / (n+4*k+1).

A025184 a(n) = T(2n,n), where T is the array defined in A025177.

Original entry on oeis.org

1, 1, 7, 35, 189, 1038, 5797, 32747, 186615, 1070762, 6177698, 35802935, 208279007, 1215507450, 7113090285, 41724381765, 245258504925, 1444292029818, 8519114704870, 50323176446818, 297654524450998

Offset: 0

Clark Kimberling

nonn

A025184 := proc(n)
        A025177(2*n,n) ;
end proc: # R. J. Mathar, Jul 28 2016

a(n) = (2*(3*n+2)*(3*n+1)*A006605(n)-(19*n+4)*(2*n-1)*A006605(n-1))/(13*n+4) for n>0. [Mark van Hoeij, Jul 02 2010]

Conjecture: 33*n*(3*n-1)*(3*n-2)*a(n) +11*(2047*n^3-10725*n^2+17192*n-8520)*a(n-1) +9*(-4397*n^3-10169*n^2+110500*n-145368)*a(n-2) -54*(2*n-5)*(5353*n^2-33313*n+53904)*a(n-3) -115668*(2*n-5)*(2*n-7)*(n-4)*a(n-4)=0. - R. J. Mathar, Feb 25 2015

cp's OEIS Frontend

A143927 G.f. satisfies: A(x) = (1 + x*A(x) + x^2*A(x)^2)^2.

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A365183 G.f. satisfies A(x) = 1 + x*A(x)^4*(1 + x*A(x)^4).

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A255673 Coefficients of A(x), which satisfies: A(x) = 1 + x*A(x)^3 + x^2*A(x)^6.

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A365189 G.f. satisfies A(x) = 1 + x*A(x)^5*(1 + x*A(x)^5).

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A143926 G.f. satisfies: A(x) = 1 + x*A(x)*A(-x) + x^2*A(x)^2*A(-x)^2.

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A026302 a(n) = number of (s(0), s(1), ..., s(n)) such that s(i) is a nonnegative integer and |s(i) - s(i-1)| <= 1 for i = 1,2,...,n, s(0) = 0, s(2n) = n. Also a(n) = T(2n,n), where T is the array in A026300.

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A365268 G.f. satisfies A(x) = 1 + x*A(x)^2*(1 + x^3*A(x)^2).

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A378292 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,0) = 0^n and T(n,k) = k * Sum_{r=0..n} binomial(2*n+k,r) * binomial(r,n-r)/(2*n+k) for k > 0.

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A143927 G.f. satisfies: A(x) = (1 + xA(x) + x^2A(x)^2)^2.

A365183 G.f. satisfies A(x) = 1 + xA(x)^4(1 + x*A(x)^4).

A255673 Coefficients of A(x), which satisfies: A(x) = 1 + xA(x)^3 + x^2A(x)^6.

A365189 G.f. satisfies A(x) = 1 + xA(x)^5(1 + x*A(x)^5).

A143926 G.f. satisfies: A(x) = 1 + xA(x)A(-x) + x^2A(x)^2A(-x)^2.

A365268 G.f. satisfies A(x) = 1 + xA(x)^2(1 + x^3*A(x)^2).

A378292 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,0) = 0^n and T(n,k) = k * Sum_{r=0..n} binomial(2n+k,r) binomial(r,n-r)/(2*n+k) for k > 0.

A364477 G.f. satisfies A(x) = 1 + xA(x)^2 + x^2A(x)^7.