A333481 -id:A333481 - cp's OEIS frontend

A080247 Formal inverse of triangle A080246. Unsigned version of A080245.

1, 2, 1, 6, 4, 1, 22, 16, 6, 1, 90, 68, 30, 8, 1, 394, 304, 146, 48, 10, 1, 1806, 1412, 714, 264, 70, 12, 1, 8558, 6752, 3534, 1408, 430, 96, 14, 1, 41586, 33028, 17718, 7432, 2490, 652, 126, 16, 1, 206098

Offset: 0

Paul Barry, Feb 15 2003

Row sums are little Schroeder numbers A001003. Diagonal sums are generalized Fibonacci numbers A006603. Columns include A006318, A006319, A006320, A006321.
T(n,k) is the number of dissections of a convex (n+3)-gon by nonintersecting diagonals with exactly k diagonals emanating from a fixed vertex. Example: T(2,1)=4 because the dissections of the convex pentagon ABCDE having exactly one diagonal emanating from the vertex A are: {AC}, {AD}, {AC,EC} and {AD,BD}. - Emeric Deutsch, May 31 2004
For more triangle sums, see A180662, see the Schroeder triangle A033877 which is the mirror of this triangle. - Johannes W. Meijer, Jul 15 2013

			Triangle starts:
[0]    1
[1]    2,    1
[2]    6,    4,   1
[3]   22,   16,   6,   1
[4]   90,   68,  30,   8,  1
[5]  394,  304, 146,  48, 10,  1
[6] 1806, 1412, 714, 264, 70, 12, 1
...
From _Gary W. Adamson_, Jul 25 2011: (Start)
n-th row = top row of M^n, M = the following infinite square production matrix:
  2, 1, 0, 0, 0, ...
  2, 2, 1, 0, 0, ...
  2, 2, 2, 1, 0, ...
  2, 2, 2, 2, 1, ...
  ... (End)

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened)
Paul Barry, Laurent Biorthogonal Polynomials and Riordan Arrays, arXiv preprint arXiv:1311.2292 [math.CA], 2013.
Paul Barry, Generalized Catalan Numbers Associated with a Family of Pascal-like Triangles, J. Int. Seq., Vol. 22 (2019), Article 19.5.8.
Paul Barry, Notes on Riordan arrays and lattice paths, arXiv:2504.09719 [math.CO], 2025. See pp. 21, 29.
James East and Nicholas Ham, Lattice paths and submonoids of Z^2, arXiv:1811.05735 [math.CO], 2018.
P. Flajolet and M. Noy, Analytic combinatorics of noncrossing configurations, Discrete Math. 204 (1999), 203-229.
Shishuo Fu and Yaling Wang, Bijective recurrences concerning two Schröder triangles, arXiv:1908.03912 [math.CO], 2019.
W.-j. Woan, The Lagrange inversion formula and divisibility properties, JIS 10 (2007) 07.7.8, example 5.

Cf. A000007, A033877 (mirror), A084938.

A080247:=(n,k)->(k+1)*add(binomial(n+1,k+j+1)*binomial(n+j,j),j=0..n-k)/(n+1):
seq(seq(A080247(n,k),k=0..n),n=0..9);

Clear[w] w[n_, k_] /; k < 0 || k > n := 0 w[0,0]=1 ; w[n_, k_] /; 0 <= k <= n && !n == k == 0 := w[n, k] = w[n-1,k-1] + w[n-1,k] + w[n,k+1] Table[w[n,k],{n,0,10},{k,0,n}] (* David Callan, Jul 03 2006 *)
T[n_, k_] := Binomial[n, k] Hypergeometric2F1[k - n, n + 1, k + 2, -1];
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Peter Luschny, Jan 08 2018 *)

T(n,k):=((k+1)*sum(2^m*binomial(n+1,m)*binomial(n-k-1,n-k-m),m,0,n-k))/(n+1); /* Vladimir Kruchinin, Jan 10 2022 */

def A080247_row(n):
    @cached_function
    def prec(n, k):
        if k==n: return 1
        if k==0: return 0
        return prec(n-1,k-1)-2*sum(prec(n,k+i-1) for i in (2..n-k+1))
    return [(-1)^(n-k)*prec(n, k) for k in (1..n)]
for n in (1..10): print(A080247_row(n)) # Peter Luschny, Mar 16 2016

G.f.: 2/(2+y*x-y+y*(x^2-6*x+1)^(1/2))/y/x. - Vladeta Jovovic, Feb 16 2003
Essentially same triangle as triangle T(n,k), n > 0 and k > 0, read by rows; given by [0, 2, 1, 2, 1, 2, 1, 2, 1, 2, ...] DELTA A000007 where DELTA is Deléham's operator defined in A084938.
T(n, k) = T(n-1, k-1) + 2*Sum_{j>=0} T(n-1, k+j) with T(0, 0) = 1 and T(n, k)=0 if k < 0. - Philippe Deléham, Jan 19 2004
T(n, k) = (k+1)*Sum_{j=0..n-k} (binomial(n+1, k+j+1)*binomial(n+j, j))/(n+1). - Emeric Deutsch, May 31 2004
Recurrence: T(0,0)=1; T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n,k+1). - David Callan, Jul 03 2006
T(n, k) = binomial(n, k)*hypergeom([k - n, n + 1], [k + 2], -1). - Peter Luschny, Jan 08 2018
T(n,k) = (k+1)/(n+1)*Sum_{m=0..n-k} 2^m*binomial(n+1,m)*binomial(n-k-1,n-k-m). - Vladimir Kruchinin, Jan 10 2022
From Peter Bala, Sep 16 2024: (Start)
Riordan array (S(x), x*S(x)), where S(x) = (1 - x - sqrt(1 - 6*x + x^2))/(2*x) is the g.f. of the large Schröder numbers A006318.
For integer m and n >= 1, (m + 2)*[x^n] S(x)^(m*n) = m*[x^n] (1/S(-x))^((m+2)*n). For cases of this identity see A103885 (m = 1), A333481 (m = 2) and A370102 (m = 3). (End)

A241023 Central terms of the triangle in A102413.

Original entry on oeis.org

1, 4, 16, 76, 384, 2004, 10672, 57628, 314368, 1728292, 9560016, 53144172, 296642688, 1661529588, 9333781872, 52566230076, 296697618432, 1677889961028, 9505151782288, 53928746972812, 306393243712384, 1742920028025364, 9925790375394096, 56584659163097436

Offset: 0

Reinhard Zumkeller, Apr 15 2014

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Cf. A001850, A006318, A047781, A102413, A333481.

Haskell
```
a241023 n = a102413 (2 * n) n
```

a[0] = 1; a[n_] := 4 Hypergeometric2F1[1 - n, n + 1, 1, -1];
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Jun 28 2019 *)

a(n) = 4 * A047781(n).
a(n) = A102413(2*n,n).
a(n) = 2*Hyper2F1([-n, n], [1], -1) for n>0. - Peter Luschny, Aug 02 2014
D-finite g.f. = (1+x)/sqrt(1-6*x+x^2), pairwise sums of A001850. - R. J. Mathar, Jan 15 2020
From Peter Bala, Apr 16 2024: (Start)
a(n) = Sum_{k = 0..n} (-1)^(n-k)*(2^k)*binomial(2*k, k)*binomial(n+k-1, n-k).
a(n) = (-1)^(n+1) * 4*n * hypergeom([n+1, -n+1], [2], 2).
n*(2*n - 3)*a(n) =  4*(3*n^2 - 6*n + 2)*a(n-1) - (2*n - 1)*(n - 2)*a(n-2) with a(0) = 1 and a(1) = 4.
O.g.f.: Sum_{n >= 0} (2^n)*binomial(2*n,n)*x^n/(1 + x)^(2*n) = 1 + 4*x + 16*x^2 + 76*x^3 + 384*x^4 + .... (End)
From Peter Bala, Sep 18 2024: (Start)
a(n) = [x^n] 1/S(-x)^(2*n), where S(x) = (1 - x - sqrt(1 - 6*x + x^2))/(2*x) is the o.g.f. of the large Schröder numbers A006318. Cf. A333481.
The Gauss congruences hold: a(n*p^r) == a(n*p^(r-1)) (mod p^r) for all primes p and positive integers n and r.
Conjecture: the supercongruences a(n*p^r) == a(n*p^(r-1)) (mod p^(3*r)) hold for all primes p >= 5 and positive integers n and r. (End)

A333482 a(n) = [x^(2n)] S(x)^n, where S(x) = (1 - x - sqrt(1 - 6x + x^2))/(2*x) is the o.g.f. of the large Schröder numbers A006318.

Original entry on oeis.org

1, 6, 304, 17718, 1093760, 69690006, 4530426640, 298634382374, 19886739416064, 1334658881073894, 90125657301992304, 6116315760393531094, 416791616968522726784, 28500344434239455360758, 1954614576511349850157392, 134392738169746273774331718, 9260873342398000417556078592

Offset: 0

Peter Bala, Mar 24 2020

Compare with the sequence A103885(n) = [x^n] S(x)^n. See also A333481.
The Gauss congruences a(n*p^k) == a(n*p^(k-1)) ( mod p^k ) hold for all prime p and positive integers n and k.
We conjecture that the stronger supercongruences a(n*p^k) == a(n*p^(k-1)) ( mod p^(3*k) ) hold for prime p >= 5 and positive integers n and k.
More generally, we conjecture that for any positive integer a and any integer b, the sequence u(a,b;n) := [x^(a*n)] S(x)^(b*n) also satisfies the above congruences.

			Examples of congruences:
a(17) - a(1) = 639400846289617183203551941830 - 6 = (2^6)*3*(17^3)* 677836910460361523003969 == 0 ( mod 17^3 ).
a(2*7) - a(2) = 1954614576511349850157392 - 304 = (2^5)*(7^4)*12408377* 2050236754217 == 0 ( mod 7^3 ).
a(5^2) - a(5) = 346904370487885277935868635823142219775940006 - 69690006 = (2^4)*(5^8)*911*39097145981*1558354721574649484551883 == 0 ( mod 5^6 )

Cf. A006318, A103885, A333481.

[1, seq((1/3)*add(binomial(3*n,k)*binomial(5*n-k-1,3*n-1), k = 0..3*n), n = 1..25)];
# alternative program
S := x -> (1/2)*(1-x-sqrt(1-6*x+x^2))/x:
G := (x, n) -> series(S(x)^n, x, 82):
seq(coeff(G(x, n), x, 2*n), n = 0..25);

Join[{1}, Table[Binomial[5*n-1, 3*n-1] * Hypergeometric2F1[-3*n, -2*n, 1 - 5*n, -1]/3, {n,1,20}]] (* Vaclav Kotesovec, Mar 28 2020 *)

a(n) = (1/3) * Sum_{k = 0..2*n} C(3*n,k)*C(5*n-k-1,3*n-1) for n >= 1.
P-recursive: P(n)*s(n+1) = 2*(1023893*n^8 - 1278327*n^6 + 474507*n^4 - 57533*n^2 + 1620)*s(n) + P(-n)*s(n-1), where P(n) = 3*(n + 1)*(2*n + 1)*(3*n + 1)*(3*n + 2)*(533*n^4 - 1066*n^3 + 736*n^2 - 203*n + 18).
a(n) ~ (1921 + 533*sqrt(13))^n / (13^(1/4) * sqrt(Pi*n) * 2^(n+1) * 3^(3*n + 1/2)). - Vaclav Kotesovec, Mar 28 2020

cp's OEIS Frontend

A080247 Formal inverse of triangle A080246. Unsigned version of A080245.

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A241023 Central terms of the triangle in A102413.

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A333482 a(n) = [x^(2n)] S(x)^n, where S(x) = (1 - x - sqrt(1 - 6x + x^2))/(2*x) is the o.g.f. of the large Schröder numbers A006318.

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cp's OEIS Frontend

A080247 Formal inverse of triangle A080246. Unsigned version of A080245.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Maxima

Sage

Formula

A241023 Central terms of the triangle in A102413.

Views

Author

Keywords

Links

Crossrefs

Programs

Haskell

Mathematica

Formula

A333482 a(n) = [x^(2*n)] S(x)^n, where S(x) = (1 - x - sqrt(1 - 6*x + x^2))/(2*x) is the o.g.f. of the large Schröder numbers A006318.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

Formula

A333482 a(n) = [x^(2n)] S(x)^n, where S(x) = (1 - x - sqrt(1 - 6x + x^2))/(2*x) is the o.g.f. of the large Schröder numbers A006318.