A121576 -id:A121576 - cp's OEIS frontend

A190736 Diagonal sums of the Riordan matrix A121576.

1, 2, 7, 29, 139, 731, 4096, 24005, 145420, 903503, 5726290, 36878978, 240663403, 1587928511, 10575884599, 71005972250, 480071241463, 3265685620913, 22335284505529, 153496543690226, 1059443187603955, 7340794592800628, 51042913856490028

Offset: 0

Emanuele Munarini, May 18 2011

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A121576.

m:=25; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!((4-5*x-2*x^2-(2+x)*Sqrt(1-8*x+4*x^2))/(2*(1-x+2*x^2+x^3)))); // G. C. Greubel, Apr 23 2018

CoefficientList[Series[(4-5x-2x^2-(2+x)Sqrt[1-8x+4x^2])/(2(1-x+2x^2 +x^3) ),{x,0,22}],x]

x='x+O('x^30); Vec((4-5*x-2*x^2-(2+x)*sqrt(1-8*x+4*x^2))/(2*(1-x+2*x^2+x^3))) \\ G. C. Greubel, Apr 23 2018

a(n) = [x^n](1-2*x-2*x^2)*(1+2*x)^(n+1)/((1+2*x-x^2+x^3)(1-x)^(n+1)).
G.f.: (4-5*x-2*x^2-(2+x)*sqrt(1-8*x+4*x^2))/(2*(1-x+2*x^2+x^3)).
Recurrence: 0 = 6*(n^2+17*n+72)*a(n+9) - (35*n^2+577*n+2376)*a(n+8) - (81*n^2+835*n+1856)*a(n+7) + (101*n^2+1017*n+2164)*a(n+6) - 2*(151*n^2+1883*n+5970)*a(n+5) - 2*(33*n^2+458*n+1528)*a(n+4) + (47*n^2+567*n+1564)*a(n+3) - 2*(7*n^2-16*n-120)*a(n+2) + 4*(3*n^2+8*n+4)*a(n+1) + 8*(n^2+3*n+2)*a(n).
Conjecture: n*(11*n-35)*a(n) + 3*(-33*n^2+149*n-136)*a(n-1) +2*(77*n^2-377*n+396)*a(n-2) +(-209*n^2+1061*n-1200)*a(n-3) +12*(-11*n+30)*a(n-4) +4*(11*n-24)*(n-4)*a(n-5)=0. - R. J. Mathar, Jul 24 2012

A190734 Central coefficients of the Riordan matrix A121576.

Original entry on oeis.org

1, 5, 51, 618, 8043, 108753, 1505652, 21181632, 301445115, 4327546335, 62550664827, 909065484726, 13271032349844, 194464001709708, 2858582670411768, 42135099729748512, 622533141544186779, 9216768941364987195, 136706015987753749593

Offset: 0

Emanuele Munarini, May 18 2011

nonn

Cf. A121576.

Sum[Binomial[2n,i]Binomial[3n-i,2n](4-9i+3i^2-12(i-1)n+8n^2)/((2n-i+2)(2n-i+1))2^i,{i,0,n}]/2

makelist(sum(binomial(2*n,i)*binomial(3*n-i,2*n)*(4-9*i+3*i^2-12*(i-1)*n+8*n^2)/((2*n-i+2)*(2*n-i+1))*2^i,i,0,n)/2,n,0,12);

a(n) = [x^n](1-2*x-2*x^2)*(1+2*x)^(2*n)/(1-x)^(2*n+1).

a(n) = sum(i=0..n, binomial(2*n,i) * binomial(3*n-i,2*n) * (8*n^2-12*(i-1)*n+3*i^2-9*i+4) / ((2*n-i+2)*(2*n-i+1))*2^i)/2

A047891 Number of planar rooted trees with n nodes and tricolored end nodes.

Original entry on oeis.org

1, 3, 12, 57, 300, 1686, 9912, 60213, 374988, 2381322, 15361896, 100389306, 663180024, 4421490924, 29712558576, 201046204173, 1368578002188, 9366084668802, 64403308499592, 444739795023054, 3082969991029800

Offset: 1

Louis Shapiro

Essentially the same as A025231.
Also number of lattice paths from (0,0) to (n-1,n-1), with steps (1,0),(0,1) and (1,1), that never rise above the line y=x and the steps (1,1) are colored red or blue. - Emeric Deutsch, May 28 2003
The Hankel transform (see A001906 for definition) of this sequence forms A049656(n+1) = [1, 3, 27, 729, 59049, 14348907, ...]. - Philippe Deléham, Aug 29 2006
With a(0)=0, this is the series reversion of x(1-x)/(1+2x). - Paul Barry, Oct 18 2009
Row sums of the Riordan matrix A121576. - Emanuele Munarini, May 18 2011

			G.f. = x + 3*x^2 + 12*x^3 + 57*x^4 + 300*x^5 + 1686*x^6 + 9912*x^7 + ...

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 = 1..200
Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
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 and A. Hennessy, A Note on Narayana Triangles and Related Polynomials, Riordan Arrays, and MIMO Capacity Calculations, J. Int. Seq. 14 (2011), Article 11.3.8.
Veronica Bitonti, Bishal Deb, and Alan D. Sokal, Thron-type continued fractions (T-fractions) for some classes of increasing trees, arXiv:2412.10214 [math.CO], 2024. See p. 58.
Zhi Chen and Hao Pan, Identities involving weighted Catalan-Schroder and Motzkin Paths, arXiv:1608.02448 [math.CO] (2016), eq. (1.13), a=3, b=1.
Shishuo Fu and Yaling Wang, Bijective recurrences concerning two Schröder triangles, arXiv:1908.03912 [math.CO], 2019.
Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.
Luis Verde-Star, A Matrix Approach to Generalized Delannoy and Schröder Arrays, J. Int. Seq., Vol. 24 (2021), Article 21.4.1.
Eric Weisstein's MathWorld, Legendre Polynomial.
Index entries for sequences related to rooted trees

Cf. A006318, A121576, A054872.

Q:=Rationals(); R:=PowerSeriesRing(Q, 40); Coefficients(R!((1-2*x-Sqrt(1-8*x+4*x^2))/(2*x))); // G. C. Greubel, Feb 10 2018

A047891_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;
for w from 1 to n do a[w] := 3*a[w-1]+add(a[j]*a[w-j-1], j=1..w-1) od; convert(a,list)end: A047891_list(20); # Peter Luschny, May 19 2011

CoefficientList[Series[(1-2x-Sqrt[1-8x+4x^2])/(2x),{x,0,100}],x] (* Emanuele Munarini, May 18 2011 *)
a[ n_] := SeriesCoefficient[(1 - 2 x - Sqrt[1 - 8 x + 4 x^2]) / 2, {x, 0, n}]; (* Michael Somos, Apr 10 2014 *)
Table[2^(n-1) (LegendreP[n, 2] - LegendreP[n-2, 2])/(2n-1), {n, 1, 20}] (* Vladimir Reshetnikov, Nov 01 2015 *)
Table[3 Hypergeometric2F1[1-n, 2-n, 2, 3] - 2 KroneckerDelta[n-1], {n, 1, 20}] (* Vladimir Reshetnikov, Nov 01 2015 *)

makelist(sum(binomial(n,k)*binomial(2*n-k+1,n+1)*(2*n^2-6*(k-1)*n+3*k^2-9*k+4)/((n-k+2)*(n-k+1))*2^k,k,0,n)/2,n,0,24); /* Emanuele Munarini, May 18 2011 */

a(n)=if(n<2,n==1,n--;sum(k=0,n,3^k*binomial(n,k)*binomial(n,k-1))/n)

x='x+O('x^100); Vec((1-2*x-sqrt(1-8*x+4*x^2))/2) \\ Altug Alkan, Nov 02 2015

G.f.: (1 - 2*x - sqrt(1 - 8*x + 4*x^2))/2.
For n>0, a(n+1) = (1/n)*Sum_{k=0..n} 3^k*C(n, k)*C(n, k-1) - Benoit Cloitre, May 10 2003
a(1)=1, a(n) = 2*a(n-1) + Sum_{i=1..(n-1)} a(i)*a(n-i). - Benoit Cloitre, Mar 16 2004
The Hankel transform (see A001906 for definition) of this sequence form A049656(n+1)= [1, 3, 27, 729, 59049, 14348907, ...]. - Philippe Deléham, Aug 29 2006
2*a(n) = A054872(n+1). - Philippe Deléham, Aug 17 2007
From Paul Barry, Feb 01 2009: (Start)
G.f.: x/(1-2x-x/(1-2x-x/(1-2x-x/(1-2x-x/(1-... (continued fraction);
a(n+1) = Sum_{k=0..n} C(n+k,2k)*2^(n-k)*A000108(k). (End)
G.f.: x/(1-3x/(1-x/(1-3x/(1-x/(1-3x/(1-x/(1-3x/(1-... (continued fraction). - Paul Barry, Oct 18 2009
a(1) = 1, for n>=1, a(n+1) = 3*A007564(n). - Aoife Hennessy (aoife.hennessy(AT)gmail.com), Dec 02 2009
From Emanuele Munarini, May 18 2011: (Start)
a(n+1) = (Sum_{k=0..n} binomial(n,k)*binomial(2*n-k+1,n+1)*(2*n^2-6*(k-1)*n+3*k^2-9*k+4)/((n-k+2)*(n-k+1))*2^k)/2.
D-finite with recurrence: (n+2)*(n+3)*a(n+3) - 6*(n+2)^2*a(n+2) - 12*(n)^2*a(n+1) + 8*n*(n-1)*a(n) = 0. (End)
G.f.: A(x) = (1-2*x-sqrt(4*x^2-8*x+1))/2 = 1 - G(0); G(k)= 1 + 2*x - 3*x/G(k+1); (continued fraction, 1-step). - Sergei N. Gladkovskii, Jan 05 2012
G.f.: x/W(0), where W(k)= k+1 - 2*x*(k+1) - x*(k+1)*(k+2)/W(k+1); (continued fraction). - Sergei N. Gladkovskii, Aug 16 2013
From Vladimir Reshetnikov, Nov 01 2015: (Start)
a(n) = 2^(n-1)*(LegendreP_n(2) - LegendreP_{n-2}(2))/(2n-1).
a(n) = 3*hypergeom([1-n,2-n], [2], 3) - 2*0^(n-1). (End)
a(n) = 2^(n-1)*hypergeom([1-n, n], [2], -1/2). - Peter Luschny, Nov 25 2020
a(n) ~ 3^(1/4) * (1 + sqrt(3))^(2*n - 1) / (2*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Jul 31 2021
D-finite with recurrence n*a(n) +4*(-2*n+3)*a(n-1) +4*(n-3)*a(n-2)=0. - R. J. Mathar, Aug 01 2022

More terms from Christian G. Bower, Dec 11 1999

A121575 Riordan array (-sqrt(4x^2+8x+1)+2x+2, (sqrt(4x^2+8x+1)-2x-1)/2).

Original entry on oeis.org

1, -2, 1, 6, -5, 1, -24, 24, -8, 1, 114, -123, 51, -11, 1, -600, 672, -312, 87, -14, 1, 3372, -3858, 1914, -618, 132, -17, 1, -19824, 22992, -11904, 4218, -1068, 186, -20, 1, 120426, -140991, 75183, -28383, 8043, -1689, 249, -23, 1, -749976, 884112, -481704, 190347, -58398, 13929, -2508, 321, -26, 1

Offset: 0

Paul Barry, Aug 08 2006

First column is (-1)^n*A054872(n). Row sums are a signed version of A108524. Inverse of generalized Delannoy triangle A121574. Unsigned triangle is A121576.

Triangle T(n,k), 0 <= k <= n, read by rows, given by [ -2, -1, -3, -1, -3, -1, -3, -1, -3, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Aug 09 2006

			Triangle begins
     1;
    -2,    1;
     6,   -5,    1;
   -24,   24,   -8,   1;
   114, -123,   51, -11,   1;
  -600,  672, -312,  87, -14, 1;

G. C. Greubel, Rows n=0..100 of triangle, flattened

T:=Flat(List([0..9],n->List([0..n],k->(-1)^(n-k)*Sum([0..n-k],i->Binomial(n,i)*Binomial(2*n-k-i,n)*(4-9*i+3*i^2-6*(i-1)*n+2*n^2)/((n-i+2)*(n-i+1))*2^i)/2))); # Muniru A Asiru, Nov 02 2018

[[(-1)^(n-k)*(&+[ 2^j*Binomial(n,j)*Binomial(2*n-k-j, n)*(4-9*j+3*j^2-6*(j-1)*n + 2*n^2)/((n-j+2)*(n-j+1))/2: j in [0..(n-k)]]): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Nov 02 2018

Flatten[Table[(-1)^(n-k)*Sum[Binomial[n, i] Binomial[2*n-k-i, n]*(4-9*i + 3*i^2 -6*(i-1)*n + 2*n^2)/((n-i+2)*(n-i+1))*2^i, {i, 0, n-k}]/2, {n, 0, 10}, {k, 0, n}]] (* G. C. Greubel, Nov 02 2018 *)

for(n=0,10, for(k=0,n, print1((-1)^(n-k)*sum(j=0, n-k, 2^j*binomial(n,j) *binomial(2*n-k-j, n)*(4-9*j+3*j^2-6*(j-1)*n + 2*n^2)/((n-j+2)*(n-j+1)))/2, ", "))) \\ G. C. Greubel, Nov 02 2018

T(n,k) = (-1)^(n-k)*(1/2)*Sum_{i=0..n-k} binomial(n,i) * binomial(2*n-k-i,n)*(4 - 9*i + 3*i^2 - 6*(i-1)*n + 2*n^2)/((n-i+2)*(n-i+1))*2^i. - G. C. Greubel, Nov 02 2018

A133366 Triangle T(n,k)read by rows given by [3,1,3,1,3,1,3,1,3,1,3,1,...] DELTA [1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 3, 1, 12, 7, 1, 57, 43, 11, 1, 300, 262, 90, 15, 1, 1686, 1618, 667, 153, 19, 1, 9912, 10159, 4745, 1336, 232, 23, 1, 60213, 64783, 33147, 10785, 2333, 327, 27, 1, 374988, 418786, 229726, 83286, 21098, 3722, 438, 31, 1

Offset: 0

Philippe Deléham, Oct 27 2007

A121576*A007318 as infinite lower triangular matrices.

			Triangle begins:
       1;
       3,      1;
      12,      7,      1;
      57,     43,     11,     1;
     300,    262,     90,    15,     1;
    1686,   1618,    667,   153,    19,    1;
    9912,  10159,   4745,  1336,   232,   23,   1;
   60213,  64783,  33147, 10785,  2333,  327,  27,  1;
  374988, 418786, 229786, 83286, 21098, 3722, 438, 31, 1; ...

Cf. A047891, A000012, A004767.

Cf. A007318, A121576.

T(0,0)=1; T(n,k) = 0 if k < 0 or if k > n; T(n,0) = 3*T(n-1,0) + 3*T(n-1,1); T(n,k) = T(n-1,k-1) + 4*T(n-1,k) + 3*T(n-1,k+1) for k >= 1.

Sum_{k>=0} T(m,k)*T(n,k)*3^k = T(m+n,0)= A047891(m+n+1). - Philippe Deléham, Jan 24 2010

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A190736 Diagonal sums of the Riordan matrix A121576.

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A190734 Central coefficients of the Riordan matrix A121576.

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A047891 Number of planar rooted trees with n nodes and tricolored end nodes.

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A121575 Riordan array (-sqrt(4*x^2+8*x+1)+2*x+2, (sqrt(4*x^2+8*x+1)-2*x-1)/2).

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A133366 Triangle T(n,k)read by rows given by [3,1,3,1,3,1,3,1,3,1,3,1,...] DELTA [1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

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A121575 Riordan array (-sqrt(4x^2+8x+1)+2x+2, (sqrt(4x^2+8x+1)-2x-1)/2).