A008288 -id:A008288 - cp's OEIS frontend

A026000 a(n) = T(2n, n), where T is the Delannoy triangle (A008288).

1, 5, 41, 377, 3649, 36365, 369305, 3800305, 39490049, 413442773, 4354393801, 46082942185, 489658242241, 5220495115997, 55818956905529, 598318746037217, 6427269150511105, 69175175263888037, 745778857519239785, 8052432236270744665, 87063177396677721409

Offset: 0

Clark Kimberling

Even order terms in the diagonal of rational function 1/(1 - (x + y^2 + x*y^2)). - Gheorghe Coserea, Aug 31 2018

			A(x) = 1 + 5*x + 41*x^2 + 377*x^3 + 3649*x^4 + 36365*x^5 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Lin Yang, Yu-Yuan Zhang, and Sheng-Liang Yang, The halves of Delannoy matrix and Chung-Feller properties of the m-Schröder paths, Linear Alg. Appl. (2024).

Cf. A008288, A027307, A026001.

Flatten[{1,RecurrenceTable[{2*n*(2*n-1)*a[n] == (46*n^2-51*n+15)*a[n-1] - (18*n^2-82*n+85)*a[n-2] - (n-2)*(2*n-5)*a[n-3],a[1]==5,a[2]==41,a[3]==377},a,{n,20}]}] (* Vaclav Kotesovec, Oct 08 2012 *)
a[n_] :=  HypergeometricPFQ[{-n, -n, n + 1}, {1/2, 1}, 1];
Table[a[n], {n, 0, 18}] (* Peter Luschny, Mar 14 2018 *)

seq(N) = {
  my(a = vector(N)); a[1]=5; a[2]=41; a[3]=377;
  for (n=4, N,
    a[n] = (46*n^2-51*n+15)*a[n-1] - (18*n^2-82*n+85)*a[n-2] - (n-2)*(2*n-5)*a[n-3];
    a[n] /= 2*n*(2*n-1));
  concat(1, a);
};
seq(18)
\\ test: y=Ser(seq(303),'x); 0 == 4*(x^2 + 11*x - 1)*y^3 + (x + 3)*y + 1
\\ Gheorghe Coserea, Aug 31 2018

a(n) = ((2*n+3)*(n+1)*A027307(n+1)/2-(3*n+2)*n*A027307(n)) / (5*n+3) (guessed). - Mark van Hoeij, Jul 02 2010
Recurrence: 2*n*(2*n-1)*a(n) = (46*n^2-51*n+15)*a(n-1) - (18*n^2-82*n+85)*a(n-2) - (n-2)*(2*n-5)*a(n-3). - Vaclav Kotesovec, Oct 08 2012
a(n) ~ sqrt(150+70*sqrt(5))*((11+5*sqrt(5))/2)^n/(20*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 08 2012. Equivalently, a(n) ~ phi^(5*n + 2) / (2 * 5^(1/4) * sqrt(Pi*n)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Dec 08 2021
a(n) = hypergeom([-n, -n, n + 1], [1/2,  1], 1). - Peter Luschny, Mar 14 2018
From Gheorghe Coserea, Aug 31 2018:(Start)
G.f.: 1 + serreverse((-(44*x^2 + 88*x + 45) + (10*x + 9)*sqrt(20*x^2 + 44*x + 25))/(8*(x + 1)^2)).
G.f. y=A(x) satisfies:
0 = 4*(x^2 + 11*x - 1)*y^3 + (x + 3)*y + 1.
0 = 2*x*(x - 2)*(x^2 + 11*x - 1)*y'' + (5*x^3 + 8*x^2 - 87*x + 2)*y' + (x^2 - 7*x - 10)*y. (End)
From Peter Bala, Jan 20 2020: (Start)
a(n) = Sum_{k = 0..n} C(2*n, n-k) * C(2*n+k, k).
a(n) = C(2*n, n) * hypergeom([-n, 2*n+1], [n+1], -1).
n*(2*n-1)*(10*n-13)*a(n) = (220*n^3-506*n^2+334*n-63*n)*a(n-1) + (n-1)*(2*n-3)*(10*n-3)*a(n-2). (End)
From Peter Bala, Apr 15 2023: (Start)
a(n) = Sum_{k = 0..n} binomial(n, k)*binomial(2*n, k)*2^k
a(n) = (-1)^n * Sum_{k = 0..n} binomial(n, k)*binomial(2*n+k, k)*(-2)^k.
a(n) = hypergeom([-n, -2*n], [1], 2) = (-1)^n * hypergeom([-n, 2*n + 1], [1], 2). (End)

A026002 a(n) = T(n,n+2), where T = Delannoy triangle (A008288).

Original entry on oeis.org

1, 7, 41, 231, 1289, 7183, 40081, 224143, 1256465, 7059735, 39753273, 224298231, 1267854873, 7178461215, 40704778785, 231128079903, 1314016698401, 7478998203943, 42612705597769, 243025194476551, 1387226559025961, 7924982285747247, 45308321651001521

Offset: 1

Clark Kimberling

Number of U steps in all lattice paths from (0,0) to (2n,0) consisting of steps U=(1,1), D=(1,-1), H=(2,0) and never going below the x-axis (i.e., Schroeder paths). For example, a(2)=7, counting the U's in HH, UDUD, UUDD, UHD, HUD and UDH. - Emeric Deutsch, Dec 06 2003
Number of UH's in all lattice paths from (0,0) to (2n+2,0) consisting of steps U=(1,1), D=(1,-1), H=(2,0) and never going below the x-axis (i.e., Schroeder paths). For example, a(2)=7, counting the UH's, shown between parentheses, in the 22 (=A006318(3)) Schroeder paths of length 6: HHH, HHUD, HUDH, HUDUD, H(UH)D, HUUDD, (UH)DH, (UH)DUD, UUDDH, UUDDUD, (UH)HD, (UH)UDD, UUDHD, UUDUDD, U(UH)DD, UUUDDD, UDHH, UDHUD, UDUDH, UDUDUD, UD(UH)D and UDUUDD. - Emeric Deutsch, Jul 16 2005
Number of walks from (0,0) to (n+2,n) using steps from {E,N,NE}. - Shanzhen Gao, May 25 2011
Conjecture: define an infinite array to have m(n,1) = m(1,n) = n*(n-1)+1 in the first row and column, and m(i,j) = m(i,j-1) + m(i-1,j-1) + m(i-1,j); then m(n,n) = a(n). - J. M. Bergot, Apr 24 2013
+-2-diagonal of A008288 as a square array. - Shel Kaphan, Jan 07 2023

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Luca Ferrari and Emanuele Munarini, Enumeration of Edges in Some Lattices of Paths, J. Int. Seq. 17 (2014) #14.1.5.
Ricardo Gómez Aíza, Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis, arXiv:2402.16111 [math.CO], 2024. See p. 19.

Cf. A002002, A008288, A110220, A190666.

List([1..25], n-> (1/n)*Sum([0..n], k-> k*Binomial(n,k)*Binomial(n+k,k+1) )); # G. C. Greubel, Feb 13 2020

[(1/n)*(&+[k*Binomial(n, k)*Binomial(n+k, k+1): k in [0..n]]): n in [1..25]]; // G. C. Greubel, Feb 13 2020

a:=n->(1/n)*sum(k*binomial(n,k)*binomial(n+k,k+1),k=0..n): seq(a(n),n=1..25); # Emeric Deutsch

Table[SeriesCoefficient[1/2-1/(2*x)+(1-4*x+x^2)/(2*x*Sqrt[1-6*x+x^2]),{x,0,n}],{n,1,25}] (* Vaclav Kotesovec, Oct 08 2012 *)
a[n_] := Binomial[2*n, n+1]*Hypergeometric2F1[-n+1, -n-1, -2*n, -1]; Flatten[Table[a[n], {n, 1, 22}]] (* Detlef Meya, Dec 26 2023 *)

my(x='x+O('x^66)); Vec( 1/2-1/(2*x)+(1-4*x+x^2)/(2*x*sqrt(1-6*x+x^2)) ) \\ Joerg Arndt, May 04 2013

a = lambda n: hypergeometric([-n-1, -n+1], [1], 2)
[simplify(a(n)) for n in (1..25)] # Peter Luschny, Nov 19 2014

From Emeric Deutsch, Dec 06 2003: (Start)
a(n) = (1/n)*Sum_{k=0..n} k*binomial(n, k)*binomial(n+k, k+1).
G.f.: 1/2 - 1/(2*z) + (1-4*z+z^2)/(2*z*sqrt(1-6*z+z^2)). (End)
a(n) = Sum_{k=0..floor(n/2)} k*A110220(n, k). - Emeric Deutsch, Jul 16 2005
a(n) = Sum_{k=0..n} C(n, k)*C(n+2, k)*2^k. - Paul Barry, Jan 23 2006
a(n) = Jacobi_P(n, 2, 0, 3). - Paul Barry, Jan 23 2006
a(n) = (-1)^n*((2*n-1)*LegendreP(n,-3)-LegendreP(n-1,-3))/(2*n+2). - Mark van Hoeij, Oct 31 2011
Recurrence: (n+1)*(6*n-7)*a(n) = (36*n^2-23*n+7)*a(n-1) - (6*n^2-n-21)*a(n-2) + (n-3)*a(n-3). - Vaclav Kotesovec, Oct 08 2012
a(n) ~ sqrt(8+6*sqrt(2))*(3+2*sqrt(2))^n/(4*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 08 2012
a(n) = hypergeom([-n-1, -n+1], [1], 2). - Peter Luschny, Nov 19 2014
From Peter Bala, Mar 02 2017: (Start)
a(n+1) = 1/2^(n+1) * Sum_{k >= 2} 1/2^k * binomial(n + k, n)*binomial(n + k, n + 2).
(n+1)*(n-1)^2*a(n) = (2*n-1)*(3*n^2 -3*n +1)*a(n-1) - (n-2)*n^2*a(n-2) with a(1) = 1 and a(2) = 7. (End)
a(n) = A001850(n) - A006318(n). - Matthew Niemiro, Jan 31 2020
a(n) = binomial(2*n, n+1)*hypergeom([-n+1, -n-1], [-2*n], -1). - Detlef Meya, Dec 26 2023

A026003 a(n) = T([n/2],[(n+1)/2]), where T = Delannoy triangle (A008288).

Original entry on oeis.org

1, 1, 3, 5, 13, 25, 63, 129, 321, 681, 1683, 3653, 8989, 19825, 48639, 108545, 265729, 598417, 1462563, 3317445, 8097453, 18474633, 45046719, 103274625, 251595969, 579168825, 1409933619, 3256957317, 7923848253, 18359266785, 44642381823

Offset: 0

Clark Kimberling

Number of lattice paths from (0,0) to the line x=n consisting of U=(1,1), D=(1,-1) and H=(2,0) steps and never going below the x-axis (i.e. left factors of Schroeder paths); for example, a(3)=5, counting the paths UUU,UUD,UDU,HU and UH. - Emeric Deutsch, Oct 27 2002
Transform of A001405 by |A049310(n,k)|, that is, transform of central binomial coefficients C(n,floor(n/2)) by Chebyshev mapping which takes a sequence with g.f. g(x) to the sequence with g.f. (1/(1-x^2))g(x/(1-x^2)). - Paul Barry, Jul 30 2005
The Kn1p sums, p >= 1, see A180662, of the Schroeder triangle A033877 (offset 0) are all related to A026003, e.g. Kn11(n) = A026003(n), Kn12(n) = A026003(n+2) - 1, Kn13(n) = A026003(n+4) - (2*n+7), Kn14(n) = A026003(n+6) - (2*n^2+18*n+41), Kn15(n) = A026003(n+8) - (4*n^3+66*n^2+368*n+693)/3, etc.. - Johannes W. Meijer, Jul 15 2013

L. Ericksen, Lattice path combinatorics for multiple product identities, J. Stat. Plan. Infer. 140 (2010) 2213-2226 doi:10.1016/j.jspi.2010.01.017
Lin Yang and S.-L. Yang, The parametric Pascal rhombus. Fib. Q., 57:4 (2019), 337-346.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Axel Bacher, Improving the Florentine algorithms: recovering algorithms for Motzkin and Schröder paths, arXiv:1802.06030 [cs.DS], 2018.
Paul Barry, The Central Coefficients of a Family of Pascal-like Triangles and Colored Lattice Paths, J. Int. Seq., Vol. 22 (2019), Article 19.1.3.
Paul Barry, Notes on Riordan arrays and lattice paths, arXiv:2504.09719 [math.CO], 2025. See pp. 20, 29.
Li-Hua Deng, Eva Y. P. Deng and Louis W. Shapiro,The Riordan Group and Symmetric Lattice Paths, arXiv:0906.1844v1 [math.CO], 2009.

Bisections are the central Delannoy numbers A001850 and A002002 respectively.

A026003 :=n -> add(binomial(n-k, k) * binomial(n-2*k, floor((n-2*k)/2)), k=0..floor(n/2)): seq(A026003(n), n=0..30); # Johannes W. Meijer, Jul 15 2013

CoefficientList[Series[(Sqrt[(x^2-2*x-1)/(x^2+2*x-1)]-1)/2/x, {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 13 2014 *)

G.f.: (sqrt((x^2-2*x-1)/(x^2+2*x-1))-1)/2/x. - Vladeta Jovovic, Apr 27 2003
a(n) = Sum_{k=0..floor(n/2)} C(n-k, k)*C(n-2k, floor((n-2k)/2)). - Paul Barry, Jul 30 2005
From Paul Barry, Mar 01 2010: (Start)
G.f.: 1/(1-x-2x^2/(1-x^2/(1-2x^2/(1-x^2/(1-2x^2/(1-... (continued fraction),
G.f.: 1/(1-x-x^2-x^2/(1-x^2-x^2/(1-x^2-x^2/(1-x^2-x^2/(1-... (continued fraction). (End)
D-finite with recurrence (n+1)*a(n) -2*a(n-1) +6*(-n+1)*a(n-2) -2*a(n-3) +(n-3)*a(n-4)=0. - R. J. Mathar, Nov 30 2012
a(n) ~ (1+sqrt(2))^(n+1) / (2^(3/4) * sqrt(Pi*n)). - Vaclav Kotesovec, Feb 13 2014

A026001 a(n) = T(3n,n), where T = Delannoy triangle (A008288).

Original entry on oeis.org

1, 7, 85, 1159, 16641, 246047, 3707509, 56610575, 872893441, 13560999991, 211939849045, 3328419072535, 52481589415425, 830317511708367, 13174519143904245, 209559710593266719, 3340604559333629953, 53354776911196959335, 853607938952248383829, 13677336690921351929767

Offset: 0

Clark Kimberling

If the Delannoy triangle is defined by the Maple code in A008288, this is A008288(n, 3*n-2), n >= 1. - N. J. A. Sloane, Oct 29 2006

Seiichi Manyama, Table of n, a(n) for n = 0..823
Lin Yang, Yu-Yuan Zhang, and Sheng-Liang Yang, The halves of Delannoy matrix and Chung-Feller properties of the m-Schröder paths, Linear Alg. Appl. (2024).

Cf. A008288, A001850, A026000, A331329.

F := (2-t)/(3*t^2-8*t+2);  G := t*(t-1)^3/(t-2);  Ginv := RootOf(numer(G-x),t); ogf := series(eval(F, t=Ginv), x=0, 25); # Mark van Hoeij, Oct 30 2011

a[n_] := Binomial[4 n, n] Hypergeometric2F1[-3 n, -n, -4 n, -1];
Array[a, 20, 0] (* Peter Luschny, Jan 31 2020 *)

G.f.: F(G^(-1)(x)) where F = (2-t)/(3*t^2-8*t+2) and G = t*(t-1)^3/(t-2). - Mark van Hoeij, Oct 30 2011
From Peter Bala, Jan 29 2020: (Start)
a(n) = Sum_{k = 0..n}  C(n,k)*C(3*n+k,n).
a(n) = Sum_{k = 0..n}  C(n,k)*C(4*n-k,n).
a(n) = Sum_{k = 0..n}  C(3*n,n-k)*C(3*n+k,k).
a(n) = Sum_{k = 0..n} 2^k*C(n,k)*C(3*n,k).
a(n) = Sum_{k = 0..n} C(4*n-k,k)*C(4*n-2*k,n-k).
3*n*(3*n - 1)*(3*n - 2)*(70*n^2 - 189*n + 127)*a(n) = 2*(15610*n^5 - 65562*n^4 + 102255*n^3 - 72864*n^2 + 23369*n - 2640)*a(n-1) - 3*(n - 1)* (3*n - 4)*(3*n - 5)*(70*n^2 - 49*n + 8)*a(n-2) with a(0) = 1, a(1) = 7.
(End)
a(n) = binomial(4*n, n)*hypergeom([-3*n, -n], [-4*n], -1). - Peter Luschny, Jan 31 2020
a(n) ~ sqrt(1 + 13/(4*sqrt(10))) * (223 + 70*sqrt(10))^n / (sqrt(Pi*n) * 3^(3*n + 1/2)). - Vaclav Kotesovec, Feb 13 2021
D-finite with recurrence +435*n*(3*n-1)*(3*n-2)*a(n) +(-53978*n^3+43545*n^2+39923*n-35580)*a(n-1) +3*(-57648*n^3+321915*n^2-580787*n+339980)*a(n-2) +9*(1634*n^3-11365*n^2+27137*n-22546)*a(n-3) -27*(3*n-10)*(3*n-11)*(n-3)*a(n-4)=0. - R. J. Mathar, Aug 01 2022

Corrected and extended by N. J. A. Sloane, Oct 29 2006

A113139 Number triangle, equal to half of Delannoy square array A008288.

Original entry on oeis.org

1, 3, 1, 13, 5, 1, 63, 25, 7, 1, 321, 129, 41, 9, 1, 1683, 681, 231, 61, 11, 1, 8989, 3653, 1289, 377, 85, 13, 1, 48639, 19825, 7183, 2241, 575, 113, 15, 1, 265729, 108545, 40081, 13073, 3649, 833, 145, 17, 1, 1462563, 598417, 224143, 75517, 22363, 5641

Offset: 0

Paul Barry, Oct 15 2005

Row sums are A047781(n+1). Diagonal sums are A113140. Inverse is A113141.

			Triangle begins
     1;
     3,    1;
    13,    5,    1;
    63,   25,    7,   1;
   321,  129,   41,   9,  1;
  1683,  681,  231,  61, 11,  1;
  8989, 3653, 1289, 377, 85, 13, 1;
  ...
A113139 as a square array = A110171 * A008288:
  / 1   1   1   1 ... \   / 1         \ / 1 1  1  1 ...\
  | 3   5   7   9 ... |   | 2  1       || 1 3  5  7 ...|
  |13  25  41  61 ... | = | 8  4 1     || 1 5 13 25 ...|
  |63 129 231 377 ... |   |38 18 6 1   || 1 7 25 63 .. |
  |...                |   |...         || 1...         |
- _Peter Bala_, Dec 09 2015

Peter Bala, Notes on generalized Riordan arrays
Peter Bala, A 4-parameter family of embedded Riordan arrays

A001850 (column 0), A002002 (column 1), A026002 (column 2), A190666 (column 3), A047781 (row sums), A113140 (diagonal sums), A113141 (matrix inverse). Cf. A006318, A008288, A110171.

T := (n,k) -> (-1)^(n-k)*hypergeom([n+1, -n+k], [1], 2):
seq(seq(simplify(T(n,k)),k=0..n),n=0..8); # Peter Luschny, Mar 02 2017

Table[Sum[Binomial[n - k, j] Binomial[n + j, k + j], {j, 0, n}], {n, 0, 9}, {k, 0, n}] // Flatten (* Michael De Vlieger, Dec 09 2015 *)

T(n, k) = Sum_{j=0..n} C(n-k, j)*C(n+j, k+j).
T(n, k) = Sum_{j=0..n} C(n, j)*C(n-k, j-k)*2^(n-j).
From Peter Bala, Dec 09 2015: (Start)
T(n,k) = A008288(n - k, n).
O.g.f.: 2/( sqrt(x^2 - 6*x + 1)*(t*sqrt(x^2 - 6*x + 1) + t*x - t + 2) ) = 1 + (3 + t)*x + (13 + 5*t + t^2)*x^2 + ....
Riordan array (f(x), x*g(x)), where f(x) = 1/sqrt(1 - 6*x + x^2) is the o.g.f. for the central Delannoy numbers, A001850, and g(x) = 1/x* revert( x*(1 - x)/(1 + x) ) = 1 + 2*x + 6*x^2 + 22*x^3 + 90*x^4 + 394*x^5 + ... is the o.g.f. for the large Schroder numbers, A006318.
Read as a square array, this is the generalized Riordan array (f(x), g(x)) in the sense of the Bala link, which factorizes as (1 + x*g'(x)/g(x), x*g(x)) * (1/(1 - x), (1 + x)/(1 - x)) = A110171 * A008288. See the example below. (End)
T(n,k) = (-1)^(n-k)*hypergeom([n+1, -n+k], [1], 2). - Peter Luschny, Mar 02 2017
From Peter Bala, Feb 16 2020: (Start)
T(n,k) = P(n-k, k, 0, 3), where P(n, alpha, beta, x) is the n-th Jacobi polynomial with parameters alpha and beta.
T(n,k) = binomial(n,k) * hypergeom( [n + 1, k - n], [k + 1], -1 ).
The n-th row polynomial in descending powers of x is the n-th Taylor polynomial of the rational function (1 + x)^n/(1 - x)^(n+1) about 0. For example, for n = 4, (1 + x)^4/(1 - x)^5 = 1 + 9*x + 41*x^2 + 129*x^3 + 321*x^4 + O(x^5). Cf. A110171. (End)

A026933 Self-convolution of array T given by A008288.

Original entry on oeis.org

1, 2, 11, 52, 269, 1414, 7575, 41064, 224665, 1237898, 6859555, 38187164, 213408805, 1196524814, 6727323439, 37915058384, 214140178225, 1211694546194, 6867622511675, 38981807403268, 221562006394173, 1260814207833750, 7182599953332423, 40958645048598840, 233779564099963081

Offset: 0

Clark Kimberling

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Paul Barry, Extensions of Riordan Arrays and Their Applications, Mathematics (2025) Vol. 13, No. 2, 242. See p. 15.

Cf. A002203, A008288, A204061, A204062.

Table[SeriesCoefficient[1/(1+x)/Sqrt[1-6*x+x^2],{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 08 2012 *)
a[ n_]:= Sum[ SeriesCoefficient[ SeriesCoefficient[1/(1-x-y-x*y) , {x,0,n-k}] , {y, 0, k}]^2, {k, 0, n}]; (* Michael Somos, Jun 27 2017 *)
A026933[n_]:= Sum[(Binomial[n, k]*Hypergeometric2F1[-k,k-n,-n,-1])^2, {k,0,n}];
Table[A026933[n], {n, 0, 40}] (* G. C. Greubel, May 25 2021 *)

/* Sum of squares of Delannoy numbers: */
{a(n)=sum(k=0,n,polcoeff(polcoeff(1/(1-x-y-x*y +x*O(x^n)+y*O(y^k)),n-k,x),k,y)^2)} \\ Paul D. Hanna, Jan 10 2012

/* Involving squares of companion Pell numbers: */
{A002203(n)=polcoeff(2*x*(1+x)/(1-2*x-x^2+x*O(x^n)),n)}
{a(n)=polcoeff(exp(sum(k=1, n, A002203(k)^2/2*x^k/k)+x*O(x^n)), n)}
\\ Paul D. Hanna, Jan 10 2012

my(x='x+O('x^66)); Vec( 1/(1+x)/sqrt(1-6*x+x^2) ) \\ Joerg Arndt, May 04 2013

def A026933_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/((1+x)*sqrt(1-6*x+x^2)) ).list()
A026933_list(40) # G. C. Greubel, May 25 2021

a(n) = Sum_{k=0..n} D(n-k,k)^2 where D(n,k) = A008288(n,k) are the Delannoy numbers. - Paul D. Hanna, Jan 10 2012
G.f.: 1/((1+x)*sqrt(1-6*x+x^2)). - Vladeta Jovovic, May 13 2003
a(n) = (-1)^n*Sum_{k=0...n} (-1)^k*A001850(k). - Benoit Cloitre, Sep 28 2005
G.f.: exp( Sum_{n>=1} A002203(n)^2/2 * x^n/n ), where A002203 are the companion Pell numbers. - Paul D. Hanna, Jan 10 2012
Self-convolution yields A204062; self-convolution of A204061. - Paul D. Hanna, Jan 10 2012
From Vaclav Kotesovec, Oct 08 2012: (Start)
Recurrence: n*a(n) = (5*n-3)*a(n-1) + (5*n-2)*a(n-2) - (n-1)*a(n-3).
a(n) ~ sqrt(24+17*sqrt(2))*(3+2*sqrt(2))^n/(8*sqrt(Pi*n)). (End)
0 = +a(n)*(+a(n+1) -8*a(n+2) -7*a(n+3) +2*a(n+4)) +a(n+1)*(-2*a(n+1) +22*a(n+2) +20*a(n+3) -7*a(n+4)) +a(n+2)*(+30*a(n+2) +22*a(n+3) -8*a(n+4)) +a(n+3)*(-2*a(n+3) +a(n+4)) for all n in Z. - Michael Somos, Jun 27 2017

More terms from Vladeta Jovovic, May 13 2003

A026937 a(n) = Sum_{k=0..n} (k+1)*T(n, n-k), where T is given by A008288.

Original entry on oeis.org

1, 3, 10, 30, 87, 245, 676, 1836, 4925, 13079, 34446, 90090, 234227, 605865, 1560200, 4002072, 10230201, 26069995, 66251090, 167941494, 424753615, 1072057117, 2700704172, 6791746500, 17052595573, 42752015487, 107035180630, 267634562754, 668407232235, 1667467065425

Offset: 0

Clark Kimberling

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Ricardo Gómez Aíza, Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis, arXiv:2402.16111 [math.CO], 2024. See p. 23.
Index entries for linear recurrences with constant coefficients, signature (4,-2,-4,-1).

Cf. A000129, A001333, A006645, A008288, A364553.

I:=[1, 3, 10, 30]; [n le 4 select I[n] else 4*Self(n-1)-2*Self(n-2)-4*Self(n-3)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Jun 20 2012

with (combinat):seq(add(fibonacci(n,2),k=0..n)/2,n=1..27); # Zerinvary Lajos, May 25 2008

CoefficientList[Series[(1-x)/(1-2x-x^2)^2,{x,0,40}],x]  (* Harvey P. Dale, Mar 22 2011 *)
LinearRecurrence[{4,-2,-4,-1},{1,3,10,30},40] (* Vincenzo Librandi, Jun 20 2012 *)
Table[(1/2)*(n+2)*Fibonacci[n+1, 2], {n, 0, 40}] (* G. C. Greubel, May 25 2021 *)

my(x='x+O('x^40)); Vec((1-x)/(1-2*x-x^2)^2) \\ Altug Alkan, Sep 20 2018

a(n) = my(w=quadgen(8)); (n/8)*((2+w)*(1+w)^n - (w-2)*(1-w)^n); \\ Michel Marcus, Jul 31 2023

[(1/2)*(n+2)*lucas_number1(n+1,2,-1) for n in (0..40)] # G. C. Greubel, May 25 2021

G.f.: (1-x)/(1 - 2*x - x^2)^2.
a(n) = Sum_{k=0..n+1} A000129(k)*A001333(n+1-k). - Graeme McRae, Aug 03 2006 and Michel Marcus, Aug 01 2023
a(n) = A006645(n+2) - A006645(n+1). - R. J. Mathar, Jan 27 2011
a(n) = 4*a(n-1) - 2*a(n-2) - 4*a(n-3) - a(n-4). - Vincenzo Librandi, Jun 20 2012
a(n) = ((n+2)/2)*A000129(n+1). - G. C. Greubel, May 25 2021
a(n) = ((n+2)/8)*((sqrt(2) + 2)*(1 + sqrt(2))^n - (sqrt(2) - 2)*(1 - sqrt(2))^n). - Peter Luschny, Jul 31 2023
a(n) = A361732(n+2)/2. - R. J. Mathar, Jun 30 2025

A297191 Irregular triangle read by rows formed by taking every other row of the Delannoy array (A008288) regarded as a triangle.

Original entry on oeis.org

1, 1, 3, 1, 1, 7, 13, 7, 1, 1, 11, 41, 63, 41, 11, 1, 1, 15, 85, 231, 321, 231, 85, 15, 1, 1, 19, 145, 575, 1289, 1683, 1289, 575, 145, 19, 1, 1, 23, 221, 1159, 3649, 7183, 8989, 7183, 3649, 1159, 221, 23, 1, 1, 27, 313, 2047, 8361, 22363, 40081, 48639, 40081

Offset: 0

N. J. A. Sloane, Jan 10 2018

			The Delannoy triangle (A008288) begins:
  1;
  1,  1;
  1,  3,   1;
  1,  5,   5,   1;
  1,  7,  13,   7,   1;
  1,  9,  25,  25,   9,   1;
  1, 11,  41,  63,  41,  11,   1;
  1, 13,  61, 129, 129,  61,  13,   1;
  1, 15,  85, 231, 321, 231,  85,  15,  1;
  1, 17, 113, 377, 681, 681, 377, 113, 17, 1;
this irregular triangle begins:
  1;
  1,  3,   1;
  1,  7,  13,   7,    1;
  1, 11,  41,  63,   41,   11,    1;
  1, 15,  85, 231,  321,  231,   85,  15,   1;
  1, 19, 145, 575, 1289, 1683, 1289, 575, 145, 19, 1;
  ...

Rémy Sigrist, Rows n = 0..100 of triangle, flattened
Rémy Sigrist, PARI program for A297191

Cf. A000129, A008288, A297192.

A297191[n_, k_]:= (-1)^k*Hypergeometric2F1[-2*n+k, k+1, 1, 2];
Table[A297191[n, k], {n,0,12}, {k,0,2*n}]//Flatten (* G. C. Greubel, May 25 2021 *)

PARI
```
See Links section.
```

def A297191(n,k): return (-1)^k*hypergeometric([-2*n+k, k+1], [1], 2).simplify()
flatten([[A297191(n,k) for k in (0..2*n)] for n in (0..12)]) # G. C. Greubel, May 25 2021

From G. C. Greubel, May 25 2021: (Start)
T(n, k) = (-1)^k*hypergeometric2F1([-2*n+k, k+1], [1], 2) for 0 <= k <= 2*n.
T(n, k) = A008288(2*n, k).
Sum_{k=0..2*n} T(n,k) = A000129(2*n+1). (End)

More terms from Rémy Sigrist, Jan 18 2018

A193653 Q-residue of the Delannoy triangle A008288, where Q is the triangular array (t(i,j)) given by t(i,j)=1.

Original entry on oeis.org

1, 2, 6, 20, 70, 248, 882, 3140, 11182, 39824, 141834, 505148, 1799110, 6407624, 22821090, 81278516, 289477726, 1030990208, 3671926074, 13077758636, 46577128054, 165886901432, 590814960402, 2104218684068, 7494285973006, 26691295287152, 95062457807466

Offset: 0

Clark Kimberling, Aug 02 2011

For the definition of Q-residue, see A193649.

This sequence gives the number of closed walks from the two vertices having loops in the digraph defined by its adjacency matrix A = (2,1,1; 1,2,1; 1,1,0). - David Neil McGrath, Aug 22 2014

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-1,-2).

Cf. A008288, A193649.

[n le 3 select Factorial(n) else 4*Self(n-1) -Self(n-2) -2*Self(n-3): n in [1..41]]; // G. C. Greubel, May 25 2021

(* First program *)
q[n_, k_] := 1;
r[0] = 1; r[k_]:= Sum[q[k-1, i]*r[k-1-i], {i, 0, k-1}]
p[n_, k_]:= p[n, k]= If[k==0 || k==n, 1, p[n-1, k-1] + p[n-2, k-1] + p[n-1, k]];  (* A008288, Delannoy *)
v[n_]:= Sum[p[n, k]*r[n-k], {k, 0, n}];
Table[v[n], {n, 0, 16}]    (* A193653 *)
TableForm[Table[q[i, k], {i, 0, 4}, {k, 0, i}]]
Table[r[k], {k, 0, 8}]  (* 2^k *)
TableForm[Table[p[n, k], {n, 0, 4}, {k, 0, n}]]
(* Second program *)
LinearRecurrence[{4,-1,-2}, {1,2,6}, 40] (* G. C. Greubel, May 25 2021 *)

Vec((1-2*t-t^2)/(1-4*t+t^2+2*t^3) + O(t^40)) \\ Michel Marcus, Aug 23 2014

a(n) = round((34+(17-3*sqrt(17))*((3-sqrt(17))/2)^n+((3+sqrt(17))/2)^n*(17+3*sqrt(17)))/68) \\ Colin Barker, Sep 02 2016

[(1/2)*(1 + sum(binomial(n-k,k)*2^k*3^(n-2*k) for k in (0..n//2))) for n in (0..40)] # G. C. Greubel, May 25 2021

From David Neil McGrath, Aug 22 2014: (Start)
a(n) = 4*a(n-1) - a(n-2) - 2*a(n-3).
a(n-1) = (1,1) and (2,2) elements of A^(n-1) where A=(2,1,1; 1,2,1; 1,1,0) and n>1. (End)
G.f.: (1-2*t-t^2)/(1-4*t+t^2+2*t^3). - Robert Israel, Aug 22 2014
a(n) = (34 + (17-3*sqrt(17))*((3-sqrt(17))/2)^n + ((3+sqrt(17))/2)^n*(17+3*sqrt(17)))/68. - Colin Barker, Sep 02 2016
From G. C. Greubel, May 25 2021: (Start)
a(n) = (1/2)*(1 + (i*sqrt(2))^n * ChebyshevU(n, -3*i/(2*sqrt(2)))).
a(n) = (1/2)*( 1 + Sum_{j=0..floor(n/2)} binomial(n-k,k)*2^k*3^(n-2*k) ). (End)

A026934 a(n) = Sum_{k=0..n-1} T(n,k) * T(n,k+1), with T given by A008288.

Original entry on oeis.org

1, 6, 35, 196, 1093, 6090, 33991, 190152, 1066313, 5993422, 33759851, 190538380, 1077316493, 6101144722, 34603634063, 196524445840, 1117492252561, 6361505951382, 36251199646387, 206773994830164, 1180452564195797, 6744529721551450, 38563791929450071, 220652949570236760

Offset: 1

Clark Kimberling

nonn

G. C. Greubel, Table of n, a(n) for n = 1..500

Cf. A008288.

A008288[n_, k_]:= Binomial[n, k]*Hypergeometric2F1[-k, k-n, -n, -1];
A026934[n_]:= Sum[A008288[n, k]*A008288[n, k+1], {k, 0, n-1}];
Table[A026934[n], {n, 1, 40}] (* G. C. Greubel, May 25 2021 *)

@CachedFunction
def A008288(n,k): return sum(binomial(n-j, j)*binomial(n-2*j, k-j) for j in (0..k))
def A026934(n): return sum(A008288(n, k)*A008288(n, k+1) for k in (0..n-1))
[A026934(n) for n in (1..40)] # G. C. Greubel, May 25 2021

More terms from Sean A. Irvine, Oct 17 2019

cp's OEIS Frontend

A026000 a(n) = T(2n, n), where T is the Delannoy triangle (A008288).

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A026002 a(n) = T(n,n+2), where T = Delannoy triangle (A008288).

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A026003 a(n) = T([n/2],[(n+1)/2]), where T = Delannoy triangle (A008288).

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A026001 a(n) = T(3n,n), where T = Delannoy triangle (A008288).

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A113139 Number triangle, equal to half of Delannoy square array A008288.

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A026933 Self-convolution of array T given by A008288.

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A026937 a(n) = Sum_{k=0..n} (k+1)*T(n, n-k), where T is given by A008288.

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