A110165 -id:A110165 - cp's OEIS frontend

A110166 Row sums of Riordan array A110165.

1, 4, 18, 85, 410, 1999, 9807, 48304, 238570, 1180615, 5851253, 29033074, 144190943, 716652070, 3564079250, 17734184365, 88280673770, 439625873215, 2189988826125, 10912480440850, 54389237971285, 271142650382080

Offset: 0

Paul Barry, Jul 14 2005

Number of 5-ary words of length n in which the number of 1's does not exceed the number of 0's. - David Scambler, Aug 14 2012
From Peter Bala, Jan 09 2022: (Start)
Conjectures: for k >= 2, the number of k-ary words of length n such that the number of 1's <= the number of 0's is equal to the coefficient of x^n in the expansion of ( k*x + 1/(1 + x) )^n, and satisfies the recurrence u(0) = 1, u(1) = k-1 and n*u(n) = (k-2)*(2*n-1)*u(n-1) - k*(k-4)*(n-1)* u(n-2) + k^(n-1) for n >= 2.
For cases see A027306 (k = 2), A027914 (k = 3) and A032443 (k = 4). (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Row sums A110165. Cf. A026375, A032443, A246437, A242586.

seq( (1/2)*(5^n + add(binomial(n,k)*binomial(2*k,k), k = 0..n)), n = 0..30); # Peter Bala, Jan 08 2022

Table[Sum[Sum[Binomial[n,j]Binomial[2j,j+k],{j,0,n}],{k,0,n}],{n,0,25}] (* Harvey P. Dale, Dec 16 2011 *)

G.f.: (1/sqrt(1-6*x+5*x^2))/(1-(1-3*x-sqrt(1-6*x+5*x^2))/(2*x)).
a(n) = Sum_{k = 0..n} Sum_{j = 0..n} C(n, j)*C(2*j, j+k).
Recurrence: n*a(n) = (11*n-8)*a(n-1) - 5*(7*n-10)*a(n-2) + 25*(n-2)*a(n-3). - Vaclav Kotesovec, Oct 18 2012
a(n) ~ 5^n/2*(1+sqrt(5)/(2*sqrt(Pi*n))). - Vaclav Kotesovec, Oct 18 2012
From Peter Bala, Jan 08 2022: (Start)
a(n) = (1/2)*(5^n + A026375(n)) = (1/2)*(5^n + Sum_{k = 0..n} binomial(n,k) *binomial(2*k,k)).
a(n) = (1/2)*(5^n)*(1 + Sum_{k = 0..n} binomial(n,k)*binomial(2*k,k)*(-1/5)^k).
a(n) = [x^n] ( 5*x + 1/(1 + x) )^n.
a(0) = 1, a(1) = 4 and n*a(n) = 3*(2*n-1)*a(n-1) - 5*(n-1)*a(n-2) + 5^(n-1) for n >= 2.
The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for prime p and positive integers n and k.
Binomial transform of A032443. (End)

A110167 Diagonal sums of Riordan array A110165.

Original entry on oeis.org

1, 3, 12, 51, 226, 1026, 4733, 22083, 103914, 492228, 2344035, 11211210, 53817063, 259135299, 1251074772, 6053946531, 29354128498, 142584834924, 693691007267, 3379680991356, 16486985580693, 80521218046569, 393674826425462

Offset: 0

Paul Barry, Jul 14 2005

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Table[Sum[Sum[Binomial[n-k,j]*Binomial[2j,j+k],{j,0,n-k}],{k,0,Floor[n/2]}],{n,0,20}] (* Vaclav Kotesovec, Oct 18 2012 *)

G.f.: 2/sqrt(1-6*x+5*x^2)/(1+3*x+sqrt(1-6*x+5*x^2)).
a(n) = sum{k=0..floor(n/2), sum{j=0..n-k, C(n-k, j)*C(2*j, j+k)}}.
Recurrence: 3*(n+1)*(16*n-7)*a(n) = (272*n^2 + 9*n - 119)*a(n-1) - 3*(48*n^2 - 37*n - 31)*a(n-2) - 5*(n-1)*(16*n+9)*a(n-3). - Vaclav Kotesovec, Oct 18 2012
a(n) ~ 5^(n+3/2)/(8*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 18 2012

A026375 a(n) = Sum_{k=0..n} binomial(n,k)binomial(2k,k).

Original entry on oeis.org

1, 3, 11, 45, 195, 873, 3989, 18483, 86515, 408105, 1936881, 9238023, 44241261, 212601015, 1024642875, 4950790605, 23973456915, 116312293305, 565280386625, 2751474553575, 13411044301945, 65448142561035, 319756851757695

Offset: 0

Clark Kimberling

nonn

a(n) is the number of integer strings s(0),...,s(n) counted by array T in A026374 that have s(n)=0; also a(n)=T(2n,n).
Partial sums of A085362. Number of bilateral Schroeder paths (i.e., lattice paths consisting of steps U=(1,1), D=(1,-1) and H=(2,0)) from (0,0) to (2n,0) and with no H-steps at odd (positive or negative) levels. Example: a(2)=11 because we have HUD, UDH, UDUD, UUDD, UDDU, their reflections in the x-axis and HH. - Emeric Deutsch, Jan 30 2004
Largest coefficient of (1+3*x+x^2)^n; row sums of triangle in A124733. - Philippe Deléham, Oct 02 2007
Also number of paths from (0,0) to (n,0) using steps U=(1,1), H=(1,0) and D=(1,-1), the H steps come in three colors. - N-E. Fahssi, Feb 05 2008
Equals INVERT transform of A109033: (1, 2, 6, 22, 88, ...), INVERTi transform of A111966, binomial transform of A000984, and inverse Binomial transform of A081671. Convolved with A002212: (1, 3, 10, 36, ...) = A026376: (1, 6, 30, 144, ...). Equals convolution square root of A003463: (1, 6, 31, 156, 781, 3906, ...). - Gary W. Adamson, May 17 2009
Diagonal of array with rational generating function 1/(1 - (x^2 + 3*x*y + y^2)). - Gheorghe Coserea, Jul 29 2018
a(n) == 0 (mod 3) if and only if n is in A081606. - Fabio Visonà, Aug 03 2023

			G.f. = 1 + 3*x + 11*x^2 + 45*x^3 + 195*x^4 + 873*x^5 + 3989*x^6 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..1000 (terms 0..200 from Vincenzo Librandi)
Hacène Belbachir, Abdelghani Mehdaoui, and László Szalay, Diagonal Sums in the Pascal Pyramid, II: Applications, J. Int. Seq., Vol. 22 (2019), Article 19.3.5.
David Callan, A combinatorial interpretation for an identity of Barrucand, JIS 11 (2008) 08.3.4.
Shu-Chiuan Chang and Robert Shrock, Structure of the Partition Function and transfer matrices for the Potts model in a magnetic field on lattice strips, J. Stat. Phys. 137 (2009) 667.
D. E. Davenport, L. W. Shapiro and L. C. Woodson, The Double Riordan Group, The Electronic Journal of Combinatorics, 18(2) (2012), #P33. - From _N. J. A. Sloane_, May 11 2012
Isaac DeJager, Madeleine Naquin, and Frank Seidl, Colored Motzkin Paths of Higher Order, VERUM 2019.
Rui Duarte and António Guedes de Oliveira, Generating functions of lattice paths, Univ. do Porto (Portugal 2023).
Francesc Fite, Kiran S. Kedlaya, Victor Rotger and Andrew V. Sutherland, Sato-Tate distributions and Galois endomorphism modules in genus 2, arXiv preprint arXiv:1110.6638 [math.NT], 2011-2012.
Francesc Fite and Andrew V. Sutherland, Sato-Tate distributions of twists of y^2= x^5-x and y^2= x^6+1, arXiv preprint arXiv:1203.1476 [math.NT], 2012. - From _N. J. A. Sloane_, Sep 14 2012
J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.
Mathematics Stack Exchange, Proof that a(n) == 0 (mod 3) if and only if n is in A081606.
László Németh, Tetrahedron trinomial coefficient transform, arXiv:1905.13475 [math.CO], 2019.
H. D. Nguyen and D. Taggart, Mining the OEIS: Ten Experimental Conjectures, 2013. Mentions this sequence. - From _N. J. A. Sloane_, Mar 16 2014
Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.
Paveł Szabłowski, Beta distributions whose moment sequences are related to integer sequences listed in the OEIS, Contrib. Disc. Math. (2024) Vol. 19, No. 4, 85-109. See p. 96.

Column 3 of A292627. Column 1 of A110165. Central column of A272866.
Cf. A002893, A084610, A000172, A002212.
First differences are in A085362. Bisection of A026380.
m-th binomial transforms of A000984: A126869 (m = -2), A002426 (m = -1 and m = -3 for signed version), A000984 (m = 0 and m = -4 for signed version), A026375 (m = 1 and m = -5 for signed version), A081671 (m = 2 and m = -6 for signed version), A098409 (m = 3 and m = -7 for signed version), A098410 (m = 4 and m = -8 for signed version), A104454 (m = 5 and m = -9 for signed version).

List([0..25],n->Sum([0..n],k->Binomial(n,k)*Binomial(2*k,k))); # Muniru A Asiru, Jul 29 2018

a026375 n = a026374 (2 * n) n  -- Reinhard Zumkeller, Feb 22 2014

seq( add(binomial(n,k)*binomial(2*k,k), k=0..n), n=0..30 ); # Detlef Pauly (dettodet(AT)yahoo.de), Nov 08 2001
a := n -> simplify(GegenbauerC(n, -n, -3/2)):
seq(a(n), n=0..22); # Peter Luschny, May 09 2016

Table[SeriesCoefficient[1/Sqrt[1-6*x+5*x^2],{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 08 2012 *)
(* From Michael Somos, May 11 2014: (Start) *)
a[ n_] := Sum[ Binomial[n, k] Binomial[2 k, k], {k, 0, n}];
a[ n_] := If[ n < 0, 0, Hypergeometric2F1[-n, 1/2, 1, -4]];
a[ n_] := If[ n < 0, 0, Coefficient[(1 + 3 x + x^2)^n, x, n]];
a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[Exp[3 x] BesselI[0,2 x], {x, 0, n}]];
(* (End) *)

A026375(n):=coeff(expand((1+3*x+x^2)^n),x,n);
makelist(A026375(n),n,0,12); /* Emanuele Munarini, Mar 02 2011 */

{a(n) = if( n<0, 0, polcoeff( (1 + 3*x + x^2)^n, n))}; /* Michael Somos, Sep 09 2002 */

a(n)={my(v=Vec((1-x-x^2)^n)); sum(k=1,#v, v[k]^2);} \\ Joerg Arndt, Jul 06 2011

{a(n) = sum(k=0, n, 5^(n-k)*(-1)^k*binomial(n, k)*binomial(2*k, k))} \\ Seiichi Manyama, Apr 22 2019

{a(n) = sum(k=0, n\2, 3^(n-2*k)*binomial(n, 2*k)*binomial(2*k, k))} \\ Seiichi Manyama, May 04 2019

Representation by Gauss's hypergeometric function, in Maple notation: a(n)=hypergeom([ -n, 1/2 ], [ 1 ], -4). - Karol A. Penson, Apr 20 2001
This sequence is the binomial transform of A000984. - John W. Layman, Aug 11 2000; proved by Emeric Deutsch, Oct 26 2002
E.g.f.: exp(3*x)*I_0(2x), where I_0 is Bessel function. - Michael Somos, Sep 17 2002
G.f.: 1/sqrt(1-6*x+5*x^2). - Emeric Deutsch, Oct 26 2002
D-finite with recurrence: n*a(n)-3*(2*n-1)*a(n-1)+5*(n-1)*a(n-2)=0 for n > 1. - Emeric Deutsch, Jan 24 2004
From Emeric Deutsch, Jan 30 2004: (Start)
a(n) = [t^n](1+3*t+t^2)^n;
a(n) = Sum_{j=ceiling(n/2)..n} 3^(2*j-n)*binomial(n, j)*binomial(j, n-j). (End)
a(n) = A026380(2*n-1) (n>0). - Emeric Deutsch, Feb 18 2004
G.f.: 1/(1-x-2*x/(1-x/(1-x-x/(1-x/(1-x-x/(1-x/(1-x-x/(1-x... (continued fraction). - Paul Barry, Jan 06 2009
a(n) = sum of squared coefficients of (1+x-x^2)^n - see triangle A084610. - Paul D. Hanna, Jul 18 2009
a(n) = sum of squares of coefficients of (1-x-x^2)^n. - Joerg Arndt, Jul 06 2011
a(n) = (1/Pi)*Integral_{x=-2..2} ((3+x)^n/sqrt((2-x)*(2+x))) dx. - Peter Luschny, Sep 12 2011
a(n) ~ 5^(n+1/2)/(2*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 08 2012
G.f.: G(0)/(1-x), where G(k) = 1 + 4*x*(4*k+1)/( (4*k+2)*(1-x) - 2*x*(1-x)*(2*k+1)*(4*k+3)/(x*(4*k+3) + (1-x)*(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 24 2013
0 = a(n)*(+25*a(n+1) - 45*a(n+2) + 10*a(n+3)) + a(n+1)*(-15*a(n+1) + 36*a(n+2) - 9*a(n+3)) + a(n+2)*(-3*a(n+2) + a(n+3)) for all n in Z. - Michael Somos, May 11 2014
a(n) = GegenbauerC(n, -n, -3/2). - Peter Luschny, May 09 2016
a(n) = Sum_{k=0..n} 5^(n-k) * (-1)^k * binomial(n,k) * binomial(2*k,k). - Seiichi Manyama, Apr 22 2019
a(n) = Sum_{k=0..floor(n/2)} 3^(n-2*k) * binomial(n,2*k) * binomial(2*k,k). - Seiichi Manyama, May 04 2019
a(n) = (1/Pi) * Integral_{x = -1..1} (1 + 4*x^2)^n/sqrt(1 - x^2) dx = (1/Pi) * Integral_{x = -1..1} (5 - 4*x^2)^n/sqrt(1 - x^2) dx. - Peter Bala, Jan 27 2020
From Peter Bala, Jan 10 2022: (Start)
1 + x*exp(Sum_{n >= 1} a(n)*x^n/n) = 1 + x + 3*x^2 + 10*x^3 + 36*x^4 + ... is the o.g.f. of A002212.
The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for prime p and positive integers n and k. (End)
a(n) = (1/4)^n * Sum_{k=0..n} 5^k * binomial(2*k,k) * binomial(2*(n-k),n-k). - Seiichi Manyama, Aug 18 2025

Definition simplified by N. J. A. Sloane, Feb 16 2012

A110168 Riordan array ((1-x^2)/(1+3x+x^2),x/(1+3x+x^2)).

Original entry on oeis.org

1, -3, 1, 7, -6, 1, -18, 24, -9, 1, 47, -84, 50, -12, 1, -123, 275, -225, 85, -15, 1, 322, -864, 900, -468, 129, -18, 1, -843, 2639, -3339, 2219, -840, 182, -21, 1, 2207, -7896, 11756, -9528, 4610, -1368, 244, -24, 1, -5778, 23256, -39825, 38121, -22518, 8532, -2079, 315, -27, 1, 15127, -67650, 130975

Offset: 0

Paul Barry, Jul 14 2005

Inverse of A110165. Row sums are 1,-2,2,-2,... with g.f. (1-x)/(1+x). Diagonal sums are (-1)^n*A080923. Product of A110162 and inverse binomial transform (1/(1+x),x/(1+x)).

			Rows begin
1;
-3,1;
7,-6,1;
-18,24,-9,1;
47,-84,50,-12,1;
-123,275,-225,85,-15,1;

T(n,k) = T(n-1,k-1) - 3*T(n-1,k) - T(n-2,k), T(0,0) = T(1,1) = T(2,2) = 1, T(1,0) = -3, T(2,0) = 7, T(2,1) = -6, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Jan 22 2014

A272866 Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k=0 and 0<=k<=2n.

Original entry on oeis.org

1, 1, 3, 1, 1, 6, 11, 6, 1, 1, 9, 30, 45, 30, 9, 1, 1, 12, 58, 144, 195, 144, 58, 12, 1, 1, 15, 95, 330, 685, 873, 685, 330, 95, 15, 1, 1, 18, 141, 630, 1770, 3258, 3989, 3258, 1770, 630, 141, 18, 1, 1, 21, 196, 1071, 3801, 9198, 15533, 18483, 15533, 9198, 3801, 1071, 196, 21, 1

Offset: 0

Peter Luschny, May 08 2016

From R. J. Mathar, Nov 05 2021: (Start)
These are the antidiagonals of the following array with the bivariate generating function 1/(1-x^2-3*x*y-y^2):
   0     1     0     1     0     1     0     1     0     1 ...
   3     0     6     0     9     0    12     0    15     0 ...
   0    11     0    30     0    58     0    95     0   141 ...
   6     0    45     0   144     0   330     0   630     0 ...
   0    30     0   195     0   685     0  1770     0  3801 ...
   9     0   144     0   873     0  3258     0  9198     0 ...
   0    58     0   685     0  3989     0 15533     0 46928 ...
  12     0   330     0  3258     0 18483     0 74280     0 ...
   0    95     0  1770     0 15533     0 86515     0 356283 ...
  15     0   630     0  9198     0 74280     0 408105     0 ...
   0   141     0  3801     0 46928     0 356283     0 1936881 ... (End)

			                                1;
                            1,  3, 1;
                         1, 6, 11, 6, 1;
                     1, 9, 30, 45, 30, 9, 1;
              1, 12, 58, 144, 195, 144, 58, 12, 1;
         1, 15, 95, 330, 685, 873, 685, 330, 95, 15, 1;

Michael De Vlieger, Table of n, a(n) for n = 0..10200 (rows 0 <= n <= 100, flattened).
Feryal Alayont and Evan Henning, Edge Covers of Caterpillars, Cycles with Pendants, and Spider Graphs, J. Int. Seq. (2023) Vol. 26, Art. 23.9.4.
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 6.
László Németh, Tetrahedron trinomial coefficient transform, Integers (2019) 19, Article A41.

Cf. A002212, A026375, A026376, A110165.

T := (n,k) -> simplify(GegenbauerC(`if`(k

Table[If[n == 0, 1, GegenbauerC[If[k < n, k, 2 n - k], -n, -3/2]], {n, 0, 7}, {k, 0, 2 n}] // Flatten (* Michael De Vlieger, Aug 02 2019 *)

T(n,n) = A026375(n) for n>=0.
T(n,n-1) = A026376(n) for n>=1.
T(n,n+1)/n = A002212(n) for n>=1.

cp's OEIS Frontend

A110166 Row sums of Riordan array A110165.

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A110167 Diagonal sums of Riordan array A110165.

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A026375 a(n) = Sum_{k=0..n} binomial(n,k)*binomial(2*k,k).

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A110168 Riordan array ((1-x^2)/(1+3x+x^2),x/(1+3x+x^2)).

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A272866 Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k=0 and 0<=k<=2n.

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A236420 Riordan array (1/(1-3*x), (1-3*x-sqrt(1-6*x+5*x^2))/(2*x)).

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A026375 a(n) = Sum_{k=0..n} binomial(n,k)binomial(2k,k).

A236420 Riordan array (1/(1-3x), (1-3x-sqrt(1-6x+5x^2))/(2*x)).