A047781 -id:A047781 - cp's OEIS frontend

A178301 Triangle T(n,k) = binomial(n,k)*binomial(n+k+1,n+1) read by rows, 0 <= k <= n.

1, 1, 3, 1, 8, 10, 1, 15, 45, 35, 1, 24, 126, 224, 126, 1, 35, 280, 840, 1050, 462, 1, 48, 540, 2400, 4950, 4752, 1716, 1, 63, 945, 5775, 17325, 27027, 21021, 6435, 1, 80, 1540, 12320, 50050, 112112, 140140, 91520, 24310, 1, 99, 2376, 24024, 126126, 378378, 672672, 700128, 393822, 92378

Offset: 0

Alford Arnold, May 30 2010

Antidiagonal sums are given by A113682. - Johannes W. Meijer, Mar 24 2013
The rows seem to give (up to sign) the coefficients in the expansion of the integer-valued polynomial binomial(x+n,n)*binomial(x+n,n-1) in the basis made of the binomial(x+i,i). - F. Chapoton, Nov 01 2022
Chapoton's observation above is correct: the precise expansion is  binomial(x+n,n)*binomial(x+n,n-1) = Sum_{k = 0..n-1} (-1)^k*T(n-1,n-1-k)*binomial(x+2*n-1-k,2*n-1-k), as can be verified using the WZ algorithm. For example, n = 4 gives binomial(x+4,4)*binomial(x+4,3)  = 35*binomial(x+7,7) - 45*binomial(x+6,6) + 15*binomial(x+5,5) - binomial(x+4,4). - Peter Bala, Jun 24 2023

			n=0: 1;
n=1: 1,  3;
n=2: 1,  8,  10;
n=3: 1, 15,  45,   35;
n=4: 1, 24, 126,  224,   126;
n=5: 1, 35, 280,  840,  1050,   462;
n=6: 1, 48, 540, 2400,  4950,  4752,  1716;
n=7: 1, 63, 945, 5775, 17325, 27027, 21021, 6435;

David A. Corneth, Table of n, a(n) for n = 0..10010
Author?, Norm of a continuous function, dxdy.ru (in Russian).

Cf. A007318, A047781 (row sums), A178300, A182626, A123160, A132813.

A178301 := proc(n,k)
        binomial(n,k)*binomial(n+k+1,n+1) ;
end proc: # R. J. Mathar, Mar 24 2013
R := proc(n) add((-1)^(n+k)*(2*k+1)*orthopoly:-P(k,2*x+1)/(n+1), k=0..n) end:
for n from 0 to 6 do seq(coeff(R(n), x, k), k=0..n) od; # Peter Luschny, Aug 25 2021

Flatten[Table[Binomial[n,k]Binomial[n+k+1,n+1],{n,0,10},{k,0,n}]] (* Harvey P. Dale, Aug 23 2014 *)

create_list(binomial(n,k)*binomial(n+k+1,n+1),n,0,12,k,0,n); /* Emanuele Munarini, Dec 16 2016 */

R(n,x) = sum(k=0,n, (-1)^(n+k) * (2*k+1) * pollegendre(k,2*x+1)) / (n+1); \\ Max Alekseyev, Aug 25 2021

T(n,k) = A007318(n,k) * A178300(n+1,k+1).
From Peter Bala, Jun 18 2015: (Start)
n-th row polynomial R(n,x) = Sum_{k = 0..n} binomial(n,k)*binomial(n+k+1,n+1)*x^k = Sum_{k = 0..n} (-1)^(n+k)*binomial(n+1,k+1)*binomial(n+k+1,n+1)*(1 + x)^k.
Recurrence: (2*n - 1)*(n + 1)*R(n,x) = 2*(4*n^2*x + 2*n^2 - x - 1)*R(n-1,x) - (2*n + 1)(n - 1)*R(n-2,x) with R(0,x) = 1, R(1,x) = 1 + 3*x.
A182626(n) = -R(n-1,-2) for n >= 1. (End)
From Peter Bala, Jul 20 2015: (Start)
n-th row polynomial R(n,x) = Jacobi_P(n,0,1,2*x + 1).
(1 + x)*R(n,x) gives the row polynomials of A123160. (End)
G.f.: (1+x-sqrt(1-2*x+x^2-4*x*y))/(2*(1+y)*x*sqrt(1-2*x+x^2-4*x*y)). - Emanuele Munarini, Dec 16 2016
R(n,x) = Sum_{k=0..n} (-1)^(n+k)*(2*k+1)*P(k,2*x+1)/(n+1), where P(k,x) is the k-th Legendre polynomial (cf. A100258) and P(k,2*x+1) is the k-th shifted Legendre polynomial (cf. A063007). - Max Alekseyev, Jun 28 2018; corrected by Peter Bala, Aug 08 2021
Polynomial g(n,x) = R(n,-x)/(n+1) delivers the maximum of f(1)^2/(Integral_{x=0..1} f(x)^2 dx) over all polynomials f(x) with real coefficients and deg(f(x)) <= n. This maximum equals (n+1)^2. See dxdy.ru link. - Max Alekseyev, Jun 28 2018

A317060 a(n) is the number of time-dependent assembly trees satisfying the edge gluing rule for a cycle on n vertices.

Original entry on oeis.org

1, 1, 3, 14, 85, 642, 5782, 60484, 720495, 9627210, 142583430, 2318126196, 41042117558, 786002475244, 16189215818220, 356847596226840, 8381418010559225, 208967274455769810, 5511890008010697306

Offset: 1

Nick Mayers, Robert Short, Aria Dougherty, Jul 20 2018

A time-dependent assembly tree for a connected graph G=(V, E) on n vertices is a rooted tree, each node of which is label a subset U of V and a nonnegative integer i such that:
1) each internal node has at least two children,
2) there are leaves labeled (v, 0) for each vertex v in V,
3) the label on the root is (V, m) for 1 <= m <= n-1,
4) for each node (U, i) with i < m, U is the union of the {u} for the children (u, 0) of (U, i),
5) if (U, i) and (U', i') are adjacent nodes with U a subset of U', then i6) for each 0 <= i <= m, there exists a node (U, i) with U a subset of V.
A time-dependent assembly tree is said to satisfy the edge gluing rule if each internal vertex v of G has exactly two children and if U_1 and U_2 are the labels of the children of internal vertex v, then there is an edge (v_1,v_2) in the edge set of G such that v_1 is in U_1 and v_2 is in U_2.

M. Bona and A. Vince, The Number of Ways to Assemble a Graph, arXiv preprint arXiv:1204.3842 [math.CO], 2012.
A. Dougherty, N. Mayers, and R. Short, How to Build a Graph in n Days: Some Variants on Graph Assembly, arXiv preprint arXiv:1807.08079 [math.CO], 2018.

Cf. A047781, A317057, A317058, A317059.

Nest[Function[{a, n}, Append[a, Sum[(Binomial[n - j, n - 2 j] + Binomial[n - j - 1, n - 2 j]) a[[n - j]], {j, Floor[n/2]}]]][#, Length@ # + 1] &, {1, 1}, 17] (* Michael De Vlieger, Jul 26 2018 *)

lista(nn) = my(v = vector(nn)); for (n=1, nn, if (n<=2, v[n] = 1, v[n] = sum(j=1, n\2, (binomial(n-j, n-2*j)+binomial(n-j-1,n-2*j))*v[n-j]))); v; \\ Michel Marcus, Aug 08 2018

@cached_function
def TimeDepenEdgeCyc(n):
    if n==1:
        return 1
    elif n==2:
        return 1
    else:
        return sum((binomial(n-j,n-2*j)+binomial(n-j-1, n-2*j))*TimeDepenEdgeCyc(n-j) for j in range(1, (n//2)+1))
print(','.join(str(TimeDepenEdgeCyc(i)) for i in range(1, 20)))

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

A361743 Central circular Delannoy numbers: a(n) is the number of Delannoy loops on an n X n toroidal grid.

Original entry on oeis.org

1, 2, 16, 114, 768, 5010, 32016, 201698, 1257472, 7777314, 47800080, 292292946, 1779856128, 10799942322, 65336473104, 394246725570, 2373580947456, 14262064668738, 85546366040592, 512323096241714, 3063932437123840, 18300660294266322, 109183694129335056

Offset: 0

Noah Snyder, Mar 22 2023

nonn

An (n,m) Delannoy loop is an oriented unbased loop on a toroidal grid with points labeled by Z/n x Z/m composed of steps of the form (1,0), (0,1), and (1,1), and which loops around the torus exactly once in each of the x-direction and the y-direction. The central circular Delannoy numbers count the number of (n,n) Delannoy loops. This is a modification of the ordinary central Delannoy numbers A001850.

Dimensions of endomorphism algebras End(S^{{n}}) in the circular Delannoy category attached to the oligomorphic group of order-preserving self-bijections of the circle.

			When n=2 see Figure 3 of "The circular Delannoy Category".

Winston de Greef, Table of n, a(n) for n = 0..1296
Nate Harman, Andrew Snowden, and Noah Snyder, The circular Delannoy Category, arxiv: 2303.10814 [math.RT], 2023.

Circular analog of A001850.
Main diagonal of A361745.
Cf. A008288, A108666, A002003, A047781.

a[n_Integer?Positive] := Sum[k Binomial[n, k] Binomial[n, k] 2^k, {k, 1, n}]

a(n) = if(n == 0, 1, sum(k=0, n, binomial(n, k)^2*k*2^k)) \\ Winston de Greef, Mar 22 2023

from math import comb
def A361743(n): return sum(comb(n,k)**2*k<Chai Wah Wu, Mar 22 2023

a(n) = Sum_{k=0..n} binomial(n,k)^2*k*2^k for n >= 1.
a(n) = n*(D(n,n-1) + D(n-1,n-1)) = n*(D(n,n) - D(n-1,n)) for n >= 1, where D(i,j) = A008288(i,j) are the Delannoy numbers.
a(n) = 2*A108666(n) for n >= 1.
From Alois P. Heinz, Mar 22 2023: (Start)
G.f.: 1 + 2*(1-x)*x/sqrt(x^2-6*x+1)^3.
a(n) = n*A002003(n) for n >= 1.
a(n) = 2*n*A047781(n) for n >= 1. (End)
a(n) = 2*n^2*hypergeom([1 - n, 1 - n], [2], 1) for n >= 1. - Peter Luschny, Mar 22 2023

A156136 A triangle of polynomial coefficients related to Mittag-Leffler polynomials: p(x,n)=Sum[Binomial[n, k]Binomial[n - 1, n - k]2^kx^k, {k, 0, n}]/(2x).

Original entry on oeis.org

1, 2, 2, 3, 12, 4, 4, 36, 48, 8, 5, 80, 240, 160, 16, 6, 150, 800, 1200, 480, 32, 7, 252, 2100, 5600, 5040, 1344, 64, 8, 392, 4704, 19600, 31360, 18816, 3584, 128, 9, 576, 9408, 56448, 141120, 150528, 64512, 9216, 256, 10, 810, 17280, 141120, 508032

Offset: 0

Roger L. Bagula, Feb 04 2009

			1;
2, 2;
3, 12, 4;
4, 36, 48, 8;
5, 80, 240, 160, 16;
6, 150, 800, 1200, 480, 32;
7, 252, 2100, 5600, 5040, 1344, 64;
8, 392, 4704, 19600, 31360, 18816, 3584, 128;
9, 576, 9408, 56448, 141120, 150528, 64512, 9216, 256;
10, 810, 17280, 141120, 508032, 846720, 645120, 207360, 23040, 512;

Steve Roman, The Umbral Calculus, Dover Publications, New York (1984), pp. 75-76

A142983, A142978, A047781 (row sums).

Clear[t0, p, x, n, m];
p[x_, n_] = Sum[Binomial[n, k]*Binomial[n - 1, n - k]*2^k*x^k, {k, 0, n}]/(2*x);
Table[FullSimplify[ExpandAll[p[x, n]]], {n, 1, 10}];
Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 1, 10}];
Flatten[%]

p(x,n)=Sum[Binomial[n, k]*Binomial[n - 1, n - k]*2^k*x^k, {k, 0, n}]/(2*x);
p(x,n)=n Hypergeometric2F1[1 - n, 1 - n, 2, 2 x];
t(n,m)=coefficiemts(p(x,n))
T(n,m) = 2^m*A103371(n,m). - R. J. Mathar, Dec 05 2017

A199856 Triangle T(n,k), read by rows, given by (-1,3,0,0,0,0,0,0,0,0,0,...) DELTA (1,0,-1/3,1/3,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, -1, 1, -2, 1, 1, -4, 1, 2, 1, -8, 0, 4, 3, 1, -16, -4, 7, 8, 4, 1, -32, -16, 10, 19, 13, 5, 1, -64, -48, 8, 41, 37, 19, 6, 1, -128, -128, -16, 80, 96, 62, 26, 7, 1, -256, -320, -112, 136, 231, 183, 95, 34, 8, 1, -512, -768, -416, 176, 518, 501, 311, 137, 43, 9, 1

Offset: 0

Philippe Deléham, Nov 11 2011

Diagonal sums are in A199853, row sums are in A000007.

			Triangle begins :
1
-1, 1
-2, 1, 1
-4, 1, 2, 1
-8, 0, 4, 3, 1
-16, -4, 7, 8, 4, 1
-32, -16, 10, 19, 13, 5, 1
-64, -48, 8, 41, 37, 19, 6, 1
-128, -128, -16, 80, 96, 62, 26, 7, 1

Cf. A000012, A000079

T(n,k)=2*T(n-1,k)+T(n-1,k-1)-T(n-2,k-1), for n>2.
T(n,0)=-2^(n-1) for n>0.
T(2*n,n)=A047781(n) for n>0.
T(n,1)=(4-n)*2^(n-3) for n>1.
G.f.: (1-3*x+x^2*y)/(1-2*x+x^2*y-x*y). - R. J. Mathar, Aug 11 2015

A272865 Triangle read by rows, T(n,k) are covariances of inverse power traces of complex Wishart matrices with parameter c=2, for n>=1 and 1<=k<=n.

Original entry on oeis.org

4, 24, 160, 132, 936, 5700, 720, 5312, 33264, 198144, 3940, 29880, 190980, 1155600, 6823620, 21672, 167712, 1088856, 6670656, 39786120, 233908896, 119812, 941640, 6189540, 38300976, 230340740, 1363667256, 7997325700

Offset: 1

Fabio Deelan Cunden, May 08 2016

These numbers provide the covariances of power traces of the time-delay matrix when the scattering matrix belongs to the Dyson ensembles.

Relation with A047781 and A002002. See eq. (60) and (61) in Cunden et al., J. Phys. A: Math. Theor. 49, 18LT01 (2016).

			Triangle starts:
4;
24,   160;
132,  936,   5700;
720,  5312,  33264,  198144;
3940, 29880, 190980, 1155600, 6823620;

F. D. Cunden, "Statistical distribution of the Wigner-Smith time-delay matrix moments for chaotic cavities", Phys. Rev. E 91, 060102(R) (2015).
F. D. Cunden, F. Mezzadri, N. Simm and P. Vivo, "Correlators for the Wigner-Smith time-delay matrix of chaotic cavities", J. Phys. A: Math. Theor. 49, 18LT01 (2016).
F. D. Cunden, F. Mezzadri, N. O'Connell and N. Simm, "Moments of Random Matrices and Hypergeometric Orthogonal Polynomials", Commun. Math. Phys. 369, 1091-1145 (2019).

F. D. Cunden, Statistical distribution of the Wigner-Smith time-delay matrix moments for chaotic cavities, arXiv:1412.2172 [cond-mat.mes-hall], 2014-2015.
F. D. Cunden, F. Mezzadri, N. Simm and P. Vivo, Correlators for the Wigner-Smith time-delay matrix of chaotic cavities, arXiv:1601.06690 [math-ph], 2016.
F. D. Cunden, F. Mezzadri, N. O'Connell and N. Simm, Moments of Random Matrices and Hypergeometric Orthogonal Polynomials, arXiv:1805.08760 [math-ph], 2018.

Cf. A002002, A047781, A050151.

P := (n,k) -> simplify(n*hypergeom([1-k,k+1],[1],-1)*hypergeom([1-n,n+1],[2],-1)): seq(seq(4*(n*k)*(P(n,k)+P(k,n))/(n+k),k=1..n),n=1..7); # Peter Luschny, May 08 2016

Clear["Global`*"];(*Wigner-Smith Covariance*)
P[k_] := Sum[Binomial[k - 1, j] Binomial[k + j, j], {j, 0, k - 1}]
Q[k_] := Sum[Binomial[k, j + 1] Binomial[k + j, j], {j, 0, k - 1}]
a[k1_, k2_] := 4 (k1 k2)/(k1 + k2) (P[k1] Q[k2] + P[k2] Q[k1])
L = 10; Table[a[k, l], {k, 1, L}, {l, 1, k}]

G.f.: ((x*y)/(x-y)^2)*((x*y-3(x+y)+1)/(sqrt(x^2-6x+1)*sqrt(y^2-6y+1))-1).

T(n,1)/4 = A050151(n) for n>=1. - Peter Luschny, May 08 2016

A344563 T(n, k) = binomial(n - 1, k - 1) * binomial(n, k) * 2^k, T(0, 0) = 1. Triangle read by rows, T(n, k) for 0 <= k <= n.

Original entry on oeis.org

1, 0, 2, 0, 4, 4, 0, 6, 24, 8, 0, 8, 72, 96, 16, 0, 10, 160, 480, 320, 32, 0, 12, 300, 1600, 2400, 960, 64, 0, 14, 504, 4200, 11200, 10080, 2688, 128, 0, 16, 784, 9408, 39200, 62720, 37632, 7168, 256, 0, 18, 1152, 18816, 112896, 282240, 301056, 129024, 18432, 512

Offset: 0

Peter Luschny, May 30 2021

			[0] 1;
[1] 0,  2;
[2] 0,  4,    4;
[3] 0,  6,   24,     8;
[4] 0,  8,   72,    96,     16;
[5] 0, 10,  160,   480,    320,     32;
[6] 0, 12,  300,  1600,   2400,    960,     64;
[7] 0, 14,  504,  4200,  11200,  10080,   2688,    128;
[8] 0, 16,  784,  9408,  39200,  62720,  37632,   7168,   256;
[9] 0, 18, 1152, 18816, 112896, 282240, 301056, 129024, 18432, 512.

T. Amdeberhan, Power of 2 dividing a specialized Mittag-Leffler polynomial, MathOverflow.

Row sums are A002003 with a(0) = 1, cf. also A047781.

The coefficients of the associated polynomials are in A103371.

aRow := n -> seq(binomial(n-1, k-1)*binomial(n,k)*2^k, k=0..n):
seq(print(aRow(n)), n=0..9);

T[n_, k_] := Binomial[n-1, k-1] * Binomial[n, k] * 2^k;
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten

from math import comb
def T(n, k):
    return comb(n-1, k-1)*comb(n, k)*2**k if k > 0 else k**n
print([T(n, k) for n in range(10) for k in range(n+1)]) # Michael S. Branicky, May 30 2021

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A178301 Triangle T(n,k) = binomial(n,k)*binomial(n+k+1,n+1) read by rows, 0 <= k <= n.

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A098660 E.g.f. BesselI(0,2*sqrt(2)*x) + BesselI(1,2*sqrt(2)*x)/sqrt(2).

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A119976 E.g.f. exp(2x)*(Bessel_I(0,2*sqrt(2)x) + Bessel_I(1,2*sqrt(2)x)/sqrt(2)).

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A296129 Array read by antidiagonals: the number of directed elements with area n on the lattice T_{2k+1}.

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A317060 a(n) is the number of time-dependent assembly trees satisfying the edge gluing rule for a cycle on n vertices.

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A361743 Central circular Delannoy numbers: a(n) is the number of Delannoy loops on an n X n toroidal grid.

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A156136 A triangle of polynomial coefficients related to Mittag-Leffler polynomials: p(x,n)=Sum[Binomial[n, k]*Binomial[n - 1, n - k]*2^k*x^k, {k, 0, n}]/(2*x).

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A098660 E.g.f. BesselI(0,2sqrt(2)x) + BesselI(1,2sqrt(2)x)/sqrt(2).

A119976 E.g.f. exp(2x)(Bessel_I(0,2sqrt(2)x) + Bessel_I(1,2*sqrt(2)x)/sqrt(2)).

A156136 A triangle of polynomial coefficients related to Mittag-Leffler polynomials: p(x,n)=Sum[Binomial[n, k]Binomial[n - 1, n - k]2^kx^k, {k, 0, n}]/(2x).