A255673 -id:A255673 - cp's OEIS frontend

A364475 G.f. satisfies A(x) = 1 + xA(x)^3 + x^2A(x)^3.

1, 1, 4, 18, 94, 529, 3135, 19270, 121732, 785496, 5155167, 34304706, 230923653, 1569684910, 10759159000, 74281473504, 516089542684, 3605685460750, 25316226436086, 178538289189108, 1264131169628799, 8982889404251721, 64041351551534215

Offset: 0

Seiichi Manyama, Jul 26 2023

nonn

Column k=1 of A378323.

Cf. A002293, A104979, A186997, A255673, A361245, A364474, A364478.

A364475 := proc(n)
    add( binomial(3*n-3*k,k) * binomial(3*n-4*k,n-2*k)/(2*n-2*k+1),k=0..n/2) ;
end proc:
seq(A364475(n),n=0..80); # R. J. Mathar, Jul 27 2023

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

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

D-finite with recurrence 2*n*(2*n+1)*a(n) -(5*n+1)*(3*n-2)*a(n-1) +4*(-25*n^2+75*n-59) *a(n-2) +9*(-15*n^2+69*n-80)*a(n-3) -6*(3*n-8)*(3*n-10) *a(n-4)=0. - R. J. Mathar, Jul 27 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).

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).

A364474 G.f. satisfies A(x) = 1 + xA(x)^3 + x^2A(x).

Original entry on oeis.org

1, 1, 4, 16, 77, 403, 2228, 12800, 75653, 457022, 2809266, 17514200, 110480475, 703850686, 4522217364, 29268545416, 190645760149, 1248817411471, 8221323983431, 54365667330636, 360954069730636, 2405225494066647, 16080210766344354, 107828663888705292

Offset: 0

Seiichi Manyama, Jul 26 2023

nonn

Michael De Vlieger, Table of n, a(n) for n = 0..1175
Paul Barry, Notes on Riordan arrays and lattice paths, arXiv:2504.09719 [math.CO], 2025. See p. 26.

Cf. A002293, A104979, A186997, A255673, A361245, A364475, A364478.

A364474 := proc(n)
    add( binomial(3*n-5*k,k) * binomial(3*n-6*k,n-2*k)/(2*n-4*k+1),k=0..n/2) ;
end proc:
seq(A364474(n),n=0..80); # R. J. Mathar, Jul 27 2023

Table[Sum[Binomial[3*n - 5*k, k]*Binomial[3*n - 6*k, n - 2*k]/(2*n - 4*k + 1), {k, 0, Floor[n/2]}], {n, 0, 25}] (* Wesley Ivan Hurt, May 25 2024 *)

a(n) = sum(k=0, n\2, binomial(3*n-5*k, k)*binomial(3*n-6*k, n-2*k)/(2*n-4*k+1));

a(n) = Sum_{k=0..floor(n/2)} binomial(3*n-5*k,k) * binomial(3*n-6*k,n-2*k) / (2*n-4*k+1).

D-finite with recurrence 2*n*(2*n+1)*(3*n-7)*a(n) -3*(3*n-1)*(3*n-7)*(3*n-2) *a(n-1) -2*(n-3)*(18*n^2-33*n+4) *a(n-2) +2*(18*n^3-141*n^2+287*n-64) *a(n-4) -2*(n-4)*(3*n-1)*(2*n-13)*a(n-6)=0. - R. J. Mathar, Jul 27 2023

A365128 G.f. satisfies A(x) = (1 + xA(x)(1 + x*A(x)))^3.

Original entry on oeis.org

1, 3, 15, 88, 567, 3876, 27607, 202653, 1522365, 11647038, 90435804, 710855544, 5645365576, 45228648078, 365109237801, 2966862631856, 24248879149005, 199213507774365, 1644138419038500, 13625326165675698, 113336685917785332, 945931091151789808

Offset: 0

Seiichi Manyama, Aug 23 2023

nonn

Cf. A001006, A143927.

Cf. A019497, A255673.

A365128 := proc(n)
    add(binomial(3*(n+1),k) * binomial(k,n-k),k=0..n) ;
    %/(n+1) ;
end proc:
seq(A365128(n),n=0..70) ; # R. J. Mathar, Dec 04 2023

a(n, s=1, t=3) = sum(k=0, n, binomial(t*(n+1), k)*binomial(s*k, n-k))/(n+1);

If g.f. satisfies A(x) = (1 + x*A(x)*(1 + x*A(x))^s)^t, then a(n) = (1/(n+1)) * Sum_{k=0..n} binomial(t*(n+1),k) * binomial(s*k,n-k).
D-finite with recurrence 205*(5*n+6)*(5*n+2)*(5*n+3)*(5*n+4)*(n+1)*a(n) +9*(-5948592*n^5+11369145*n^4 -5182620*n^3 -351495*n^2+204302*n-6560) *a(n-1) +243*(-801282*n^5 +14391105*n^4 -55889790*n^3 +90254895*n^2 -66199848*n +18182560)*a(n-2) +6561*(3*n-5) *(3*n-4)*(93048*n^3 -579621*n^2 +1227037*n -878874)*a(n-3) +48715425*(n-3) *(3*n-4)*(3*n-7) *(3*n-5)*(3*n-8)*a(n-4)=0. - R. J. Mathar, Dec 04 2023
From Seiichi Manyama, Sep 20 2024: (Start)
G.f.: (1/x) * Series_Reversion( x / (1+x+x^2)^3 ).
G.f.: B(x)^3, where B(x) is the g.f. of A255673. (End)

A364478 G.f. satisfies A(x) = 1 + xA(x)^3 + x^2A(x)^8.

Original entry on oeis.org

1, 1, 4, 23, 154, 1124, 8675, 69626, 575243, 4859778, 41789764, 364565277, 3218581695, 28702642553, 258172627259, 2339496034381, 21337716782873, 195726876816623, 1804472496834650, 16711389876481027, 155395461519245354, 1450298253483719944

Offset: 0

Seiichi Manyama, Jul 26 2023

nonn

Cf. A002293, A104979, A186997, A255673, A361245, A364474, A364475.

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

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

A367040 G.f. satisfies A(x) = 1 + x^2 + x*A(x)^3.

Original entry on oeis.org

1, 1, 4, 15, 70, 360, 1953, 11008, 63837, 378390, 2282205, 13960890, 86411232, 540166219, 3405341160, 21625820793, 138216775785, 888371346825, 5738510504979, 37234351046835, 242567430368298, 1585979835198675, 10403866383915844, 68453912880893025

Offset: 0

Seiichi Manyama, Nov 03 2023

nonn

Cf. A176697, A367041.
Cf. A366266, A366676.
Cf. A002293, A104979, A186997, A255673, A361245, A364474, A364475, A364478.

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

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

A261440 Array of coefficients A(n,k) of the formal power series P(n,x) read by upwards antidiagonals, where P(n,x) = Sum_{k>=0} A(n,k)x^k = 1+xP(n,x)^(1n)+x^2P(n,x)^(2*n) for n >= 0.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 0, 1, 1, 3, 4, 0, 1, 1, 4, 11, 9, 0, 1, 1, 5, 21, 46, 21, 0, 1, 1, 6, 34, 127, 207, 51, 0, 1, 1, 7, 50, 268, 833, 979, 127, 0, 1, 1, 8, 69, 485, 2299, 5763, 4797, 323, 0, 1, 1, 9, 91, 794, 5130, 20838, 41401, 24138, 835, 0

Offset: 0

Werner Schulte, Aug 18 2015

The terms define the array A(n,k):
  n\k:  0  1   2    3    4     5      6      7       8        9   10  ...
    0:  1  1   1    0    0     0      0      0       0        0    0  ...
    1:  1  1   2    4    9    21     51    127     323      835  ...
    2:  1  1   3   11   46   207    979   4797   24138   123998  ...
    3:  1  1   4   21  127   833   5763  41401  305877  2309385  ...
    4:  1  1   5   34  268  2299  20838  ...
    5:  1  1   6   50  485  5130  ...
    6:  1  1   7   69  794  ...
    7:  1  1   8   91  ...
    8:  1  1   9  116  ...
    9:  1  1  10  144  ...
   10:  1  ...
  etc.
For row 1 see A001006, for row 2 see A006605, and for row 3 see A255673.
Be careful if you use the formulas for n < 0 (DIV0, signed values)!
Nevertheless, it might be interesting ...
Conjecture: The A(n,k), here n > 0, are the number of lattice paths, if
(a) length of path is k*n for the k-th term of row n,
(b) allowed steps are (1,-1), (1,-1+n) and (1,-1+2*n) for terms of row n,
(c) you start at (0,0), end at (k*n,0), and
(d) never cross the x-axis.
This is proved for row 1 (A001006) and row 2 (A006605).
Conjecture: The coefficients B(m,n,k) of the P(n,x)^m (see the formula below), m > 0 and n > 0, are the number of lattice paths, if
(a) length of path is k*n+m-1 (k-th coefficient of P(n,x)^m),
(b) allowed steps are (1,-1), (1,-1+n), and (1,-1+2*n),
(c) you start at (0,m-1), end at (k*n+m-1,0), and
(d) never cross the x-axis.
This is proved for B(1,1,k) (A001006), and B(1,2,k) (A006605). - Werner Schulte, Aug 30 2015

			The terms of the array A(n,k) read by upwards antidiagonals define the triangle T(n,m) = A(n-m,m) for 0 <= m <= n, i.e.
  1;
  1, 1;
  1, 1, 1;
  1, 1, 2,  0;
  1, 1, 3,  4,  0;
  1, 1, 4, 11,  9,  0;
  1, 1, 5, 21, 46, 21, 0;
  etc.

Cf. A001006, A006605, A255673.

A(n,k) = 1/(n*k+1)*Sum_{j=0..k} (-1)^j*binomial(n*k+1, j)*binomial(2*n*k+2-2*j, k-j) (conjectured).
The g.f. P(n,x) of row n of the array A(n,k) satisfy:
  P(n,x) = (1 + x*P(n,x)^n)^2/(1 + x*P(n,x)^(n-1)), n > 0.
  P(n,x) = P(n-1,x*P(n,x)), n > 0.
  P(n,x) = P(n-2,x*P(n,x)^2), n > 1.
  etc.
  P(n,x) = P(0,x*P(n,x)^n), n >= 0.
The coefficients B(m,n,k) of the P(n,x)^m are:
  B(m,n,k) = m/(n*k + m)*(Sum_{j=0..k} (-1)^j*binomial(n*k+m, j)* binomial(2*n*k + 2*m - 2*j, k - j)), if m > 0, and n > 0 (conjectured).
A(n,0) = A(n,1) = 1, n >= 0.
A(n,2) = n+1, n >= 0.
A(n,3) = n*(3*n + 5)/2, n >= 0.
A(n,4) = n*(8*n^2 + 18*n + 1)/3, n >= 0.
A(n,5) = n*(125*n^3 + 350*n^2 + 55*n - 26)/24, n >= 0.
P(n,x) = exp(Sum_{k>=1} 1/(n*k)*(Sum{j=0..k} (-1)^j*binomial(n*k,j)* binomial(2*n*k-2*j,k-j))) for n > 0 (conjectured). - Werner Schulte, Sep 20 2015
P(n,x/(1+x+x^2)^n) = 1+x+x^2 for n >= 0. - Werner Schulte, Oct 20 2015

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A364475 G.f. satisfies A(x) = 1 + x*A(x)^3 + x^2*A(x)^3.

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

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

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A364474 G.f. satisfies A(x) = 1 + x*A(x)^3 + x^2*A(x).

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A365128 G.f. satisfies A(x) = (1 + x*A(x)*(1 + x*A(x)))^3.

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A364478 G.f. satisfies A(x) = 1 + x*A(x)^3 + x^2*A(x)^8.

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A367040 G.f. satisfies A(x) = 1 + x^2 + x*A(x)^3.

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A261440 Array of coefficients A(n,k) of the formal power series P(n,x) read by upwards antidiagonals, where P(n,x) = Sum_{k>=0} A(n,k)*x^k = 1+x*P(n,x)^(1*n)+x^2*P(n,x)^(2*n) for n >= 0.

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A364475 G.f. satisfies A(x) = 1 + xA(x)^3 + x^2A(x)^3.

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

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

A364474 G.f. satisfies A(x) = 1 + xA(x)^3 + x^2A(x).

A365128 G.f. satisfies A(x) = (1 + xA(x)(1 + x*A(x)))^3.

A364478 G.f. satisfies A(x) = 1 + xA(x)^3 + x^2A(x)^8.

A261440 Array of coefficients A(n,k) of the formal power series P(n,x) read by upwards antidiagonals, where P(n,x) = Sum_{k>=0} A(n,k)x^k = 1+xP(n,x)^(1n)+x^2P(n,x)^(2*n) for n >= 0.