A026378 -id:A026378 - cp's OEIS frontend

A026392 T(n,[ n/2 ]), where T is the array in A026386.

1, 2, 4, 8, 17, 34, 75, 150, 339, 678, 1558, 3116, 7247, 14494, 34016, 68032, 160795, 321590, 764388, 1528776, 3650571, 7301142, 17501619, 35003238, 84179877, 168359754, 406020930, 812041860, 1963073865, 3926147730

Offset: 1

Clark Kimberling

nonn

Paul Barry, Extensions of Riordan Arrays and Their Applications, Mathematics (2025) Vol. 13, No. 2, 242. See p. 15.

Cf. A026378, A085362.

A026392 := proc(n)
    A026386(n,floor(n/2)) ;
end proc: # R. J. Mathar, Feb 10 2015

Conjecture: (n+1)*a(n) +2*(n-1)*a(n-1) +2*(-3*n+1)*a(n-2) +4*(-3*n+7)*a(n-3) +5*(n-3)*a(n-4) +10*(n-5)*a(n-5)=0. - R. J. Mathar, Feb 10 2015

Offset corrected. R. J. Mathar, Feb 10 2015

A126182 Let P be Pascal's triangle A007318 and let N be Narayana's triangle A001263, both regarded as lower triangular matrices. Sequence gives triangle obtained from P*N, read by rows.

Original entry on oeis.org

1, 2, 1, 4, 5, 1, 8, 18, 9, 1, 16, 56, 50, 14, 1, 32, 160, 220, 110, 20, 1, 64, 432, 840, 645, 210, 27, 1, 128, 1120, 2912, 3150, 1575, 364, 35, 1, 256, 2816, 9408, 13552, 9534, 3388, 588, 44, 1, 512, 6912, 28800, 53088, 49644, 24822, 6636, 900, 54, 1

Offset: 0

Emeric Deutsch, Dec 19 2006, Mar 30 2007

Also T(n,k) is number of hex trees with n edges and k left edges (0<=k<=n). A hex tree is a rooted tree where each vertex has 0, 1, or 2 children and, when only one child is present, it is either a left child, or a median child, or a right child (name due to an obvious bijection with certain tree-like polyhexes; see the Harary-Read reference). Accordingly, one can have left, vertical, or right edges.
Also (with a different offset) T(n,k) is the number of skew Dyck paths of semilength n and having k peaks (1<=k<=n). A skew Dyck path is a path in the first quadrant which begins at the origin, ends on the x-axis, consists of steps U=(1,1)(up), D=(1,-1)(down) and L=(-1,-1)(left) so that up and left steps do not overlap. The length of the path is defined to be the number of its steps. E.g., T(3,2)=5 because we have (UD)U(UD)D, (UD)U(UD)L, U(UD)D(UD), U(UD)(UD)D and U(UD)(UD)L (the peaks are shown between parentheses).
Sum of terms in row n = A002212(n+1). T(n,1) = A001793(n); T(n,2) = A006974(n-2); Sum_{k=0..n}kT(n,k) = A026379(n+1).
A126216 = N * P. - Gary W. Adamson, Nov 30 2007

			The triangle P begins
  1,
  1, 1
  1, 2, 1
  1, 3, 3, 1, ...
and T begins
  1,
  1,  1,
  1,  3,  1,
  1,  6,  6,  1,
  1, 10, 20, 10, 1, ...
The product P*T gives
   1,
   2,  1,
   4,  5,  1,
   8, 18,  9,  1,
  16, 56, 50, 14, 1, ...

E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203.
M. Dziemianczuk, Enumerations of plane trees with multiple edges and Raney lattice paths, Discrete Mathematics 337 (2014): 9-24.
F. Harary and R. C. Read, The enumeration of tree-like polyhexes, Proc. Edinburgh Math. Soc. (2) 17 (1970), 1-13.

Cf. A001263, A007318, A002212, A001793, A006974, A026379.

Cf. A128718, A026378, A126216.

T:=proc(n,k) if k=0 then 2^n elif k<=n then binomial(n+1,k)*sum(binomial(k,n-k-j)*binomial(n+1-k,j)*2^j,j=n-2*k..n-k)/(n+1) else 0 fi end: for n from 0 to 10 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form

t[n_, 0] := 2^n; t[n_, k_] := Binomial[n+1, k]*Sum[Binomial[k, n-k-j]*Binomial[n+1-k, j]*2^j, {j, n-2*k, n-k}]/(n+1); Table[t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 12 2013 *)
nmax = 10; n[x_, y_] := (1-x*(1+y) - Sqrt[(1-x*(1+y))^2 - 4*y*x^2])/(2*x); s = Series[(n[x/(1-x), y]-1)/x, {x, 0, nmax+1}, {y, 0, nmax+1}];t[n_, k_] := SeriesCoefficient[s, {x, 0, n}, {y, 0, k}]; Table[t[n, k], {n, 0, nmax}, {k, 1, n+1}] // Flatten (* Jean-François Alcover, Apr 16 2015, after Vladimir Kruchinin *)

tabl(nn) = {mP = matrix(nn, nn, n, k, binomial(n-1, k-1)); mN = matrix(nn, nn, n, k, binomial(n-1, k-1) * binomial(n, k-1) / k); mprod = mP*mN; for (n=1, nn, for (k=1, n, print1(mprod[n, k], ", ");); print(););} \\ Michel Marcus, Apr 16 2015

T(n,k) = (1/(n+1))*binomial(n+1,k)*Sum_{j=n-2k..n-k}2^j*binomial(k,n-k-j)*binomial(n+1-k,j) if 0 < k <= n; T(n,0) = 2^n.
G.f. G=G(t,z) satisfies G  = 1 + (t+2)*z*G + t*z^2*G^2.
E.g.f.: exp((t+2)*x)*BesselI_{1}(2*sqrt(t)*x)/(sqrt(t)*x). - Peter Luschny, Oct 29 2014
G.f.: N(x/(1-x),y)-1)/x, where N(x,y) is the g.f. of Narayana's triangle A001263. - Vladimir Kruchinin, Apr 06 2015.

New definition in terms of P and N from Philippe Deléham, Jun 30 2007

Edited by N. J. A. Sloane, Jul 22 2007

A171651 Triangle T, read by rows : T(n,k) = A007318(n,k)*A005773(n+1-k).

Original entry on oeis.org

1, 2, 1, 5, 4, 1, 13, 15, 6, 1, 35, 52, 30, 8, 1, 96, 175, 130, 50, 10, 1, 267, 576, 525, 260, 75, 12, 1, 750, 1869, 2016, 1225, 455, 105, 14, 1, 2123, 6000, 7476, 5376, 2450, 728, 140, 16, 1, 6046, 19107, 27000, 22428, 12096, 4410, 1092, 180, 18, 1

Offset: 0

Philippe Deléham, Dec 14 2009

			Triangle begins:
   1;
   2,   1;
   5,   4,  1;
  13,  15,  6, 1;
  35,  52, 30, 8, 1;
  ...

Alois P. Heinz, Rows n = 0..140, flattened

Cf. A097692.
Cf. A007318, A005773.
Cf. A168491, A099323, A001405, A005773, A001700, A026378, A005573, A122898.

b:= proc(u, d, t) option remember; `if`(u=0 and d=0, 1/2,
      expand(`if`(u=0, 0, b(u-1, d, 2)*`if`(t=3, x, 1))
      +`if`(d=0, 0, b(u, d-1, `if`(t=2, 3, 1)))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n+1$2, 1)):
seq(T(n), n=0..12);  # Alois P. Heinz, Apr 29 2015
# second program:
A171651:= (n, k)-> binomial(n,k)*add((-1)^(n-k-j)*binomial(n-k,j)*binomial(2*j+1,j+1),j=0..n-k): seq(print(seq(A171651(n, k), k=0..n)), n=0..9);  # Mélika Tebni, Dec 16 2023

b[u_, d_, t_] := b[u, d, t] = If[u == 0 && d == 0, 1/2, Expand[If[u == 0, 0, b[u-1, d, 2]*If[t == 3, x, 1]] + If[d == 0, 0, b[u, d-1, If[t == 2, 3, 1]]]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n+1, n+1, 1] ];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, May 21 2016, after Alois P. Heinz *)

Sum_{k, 0<=k<=n} T(n,k)*x^k = A168491(n), A099323(n), A001405(n), A005773(n+1), A001700(n), A026378(n+1), A005573(n), A122898(n) for x = -3, -2, -1, 0, 1, 2, 3, 4 respectively.

E.g.f. of column k: exp(x)*(BesselI(0,2*x)+BesselI(1,2*x))*x^k / k!. - Mélika Tebni, Dec 16 2023

Corrected by Philippe Deléham, Dec 18 2009

A026387 a(n) = number of integer strings s(0),...,s(n) counted by array T in A026386 that have s(n)=0; also a(n) = T(2n,n).

Original entry on oeis.org

2, 8, 34, 150, 678, 3116, 14494, 68032, 321590, 1528776, 7301142, 35003238, 168359754, 812041860, 3926147730, 19022666310, 92338836390, 448968093320, 2186194166950, 10659569748370, 52037098259090, 254308709196660

Offset: 0

Clark Kimberling

László Németh, Tetrahedron trinomial coefficient transform, arXiv:1905.13475 [math.CO], 2019.

Essentially the same as A085362.

Cf. A026378.

a(n) = A085362(n+1), n >= 0. - Hartmut F. W. Hoft, Jul 07 2024

A129164 Sum of pyramid weights in all skew Dyck paths of semilength n.

Original entry on oeis.org

1, 5, 22, 97, 436, 1994, 9241, 43257, 204052, 968440, 4619011, 22120630, 106300507, 512321437, 2475395302, 11986728457, 58156146652, 282640193312, 1375737276787, 6705522150972, 32724071280517, 159878425878847, 781910419686412, 3827639591654862, 18753350784435811

Offset: 1

Emeric Deutsch, Apr 03 2007

nonn

A skew Dyck path is a path in the first quadrant which begins at the origin, ends on the x-axis, consists of steps U=(1,1)(up), D=(1,-1)(down) and L=(-1,-1)(left) so that up and left steps do not overlap. The length of the path is defined to be the number of its steps. A pyramid in a skew Dyck word (path) is a factor of the form U^h D^h, h being the height of the pyramid. A pyramid in a skew Dyck word w is maximal if, as a factor in w, it is not immediately preceded by a U and immediately followed by a D. The pyramid weight of a skew Dyck path (word) is the sum of the heights of its maximal pyramids.

			a(2)=5 because the pyramid weights of the paths (UD)(UD), (UUDD) and U(UD)L are 2, 2 and 1, respectively (the maximal pyramids are shown between parentheses).

Vincenzo Librandi, Table of n, a(n) for n = 1..200
A. Denise and R. Simion, Two combinatorial statistics on Dyck paths, Discrete Math., 137, 1995, 155-176.
E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203

Cf. A026378, A129163.

G:=(1/sqrt(1-6*z+5*z^2)-1/(1-z))/2: Gser:=series(G,z=0,30): seq(coeff(Gser,z,n),n=1..26);

Rest[CoefficientList[Series[(1/Sqrt[1-6*x+5*x^2]-1/(1-x))/2, {x, 0, 20}], x]] (* Vaclav Kotesovec, Oct 20 2012 *)
a[n_] := (Hypergeometric2F1[1/2, -n, 1, -4]-1)/2; Array[a, 25] (* Jean-François Alcover, Oct 11 2016, after Vladimir Kruchinin *)

a(n):=n*sum(sum((binomial(-m+2*k-1,k-1)*binomial(n-1,k-1))/k,k,m,n), m,1,n); /* Vladimir Kruchinin, Oct 07 2011 */

a(n):=sum(binomial(n,k)*binomial(2*k,k),k,1,n)/2; /* Vladimir Kruchinin, Oct 11 2016 */

A129164 = lambda n: n*hypergeometric([1, 3/2, 1-n], [2, 2], -4)
[simplify(A129164(n)) for n in (1..25)] # Peter Luschny, Sep 16 2014

a(n) = Sum_{k=1..n} k*A129163(n,k).
Partial sums of A026378.
G.f. = [1/sqrt(1-6*z+5*z^2)-1/(1-z)]/2.
a(n) = n*Sum_(m=1..n, Sum_(k=m..n,(binomial(-m+2*k-1,k-1)*binomial(n-1,k-1))/k)). - Vladimir Kruchinin, Oct 07 2011
Recurrence: n*a(n) = (7*n-4)*a(n-1) - (11*n-14)*a(n-2) + 5*(n-2)*a(n-3). - Vaclav Kotesovec, Oct 20 2012
a(n) ~ 5^(n+1/2)/(4*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 20 2012
a(n) = n*hypergeometric([1, 3/2, 1-n], [2, 2], -4). - Peter Luschny, Sep 16 2014
a(n) = Sum_{k=1..n} (binomial(n,k)*binomial(2*k,k))/2. - Vladimir Kruchinin, Oct 11 2016

A171224 Riordan array (f(x),x*f(x)) where f(x) is the g.f. of A117641.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 3, 2, 0, 1, 11, 6, 3, 0, 1, 42, 23, 9, 4, 0, 1, 167, 90, 36, 12, 5, 0, 1, 684, 365, 144, 50, 15, 6, 0, 1, 2867, 1518, 595, 204, 65, 18, 7, 0, 1, 12240, 6441, 2511, 858, 270, 81, 21, 8, 0, 1, 53043, 27774, 10782, 3672, 1155, 342, 98, 24, 9, 0, 1

Offset: 0

Philippe Deléham, Dec 05 2009

			Triangle begins
    1;
    0,  1;
    1,  0,  1;
    3,  2,  0,  1;
   11,  6,  3,  0,  1;
   42, 23,  9,  4,  0,  1;
  167, 90, 36, 12,  5,  0,  1;
  ...
Production array begins
    0,  1;
    1,  0,  1;
    3,  1,  0,  1;
    9,  3,  1,  0,  1;
   27,  9,  3,  1,  0,  1;
   81, 27,  9,  3,  1,  0,  1;
  243, 81, 27,  9,  3,  1,  0,  1;
  ... - _Philippe Deléham_, Mar 04 2013

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

Cf. A053121, A097609, A065600.

[[((k+1)/(n+1))*(&+[3^(n-k-2*j)*Binomial(n+1,j)*Binomial(n-k-j-1, n-k-2*j): j in [0..Floor((n-k)/2)]]): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Apr 04 2019

T[n_, k_]:= (k+1)/(n+1)*Sum[3^(n-k-2*j)*Binomial[n+1,j]*Binomial[n-k-j-1, n-k-2*j], {j, 0, Floor[(n-k)/2]}]; Table[T[n, k], {n,0,10}, {k,0,n} ]//Flatten (* G. C. Greubel, Apr 04 2019 *)

T(n,k):=(k+1)/(n+1)*sum(3^(n-k-2*j)*binomial(n+1,j)*binomial(n-k-j-1,n-k-2*j),j,0,floor((n-k)/2)); /* Vladimir Kruchinin, Apr 04 2019 */

{T(n,k) = ((k+1)/(n+1))*sum(j=0, floor((n-k)/2), 3^(n-k-2*j) *binomial(n+1,j)*binomial(n-k-j-1, n-k-2*j))}; \\ G. C. Greubel, Apr 04 2019

[[((k+1)/(n+1))*sum(3^(n-k-2*j)*binomial(n+1,j)*binomial(n-k-j-1, n-k-2*j) for j in (0..floor((n-k)/2))) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Apr 04 2019

Sum_{k=0..n} T(n,k)*x^k = A117641(n), A033321(n), A007317(n+1), A002212(n+1), A026378(n+1) for x = 0, 1, 2, 3, 4 respectively.
Triangle equals B*A065600*B^(-1) = B^2*A097609*B^(-2) = B^3*A053121*B^(-3), product considered as infinite lower triangular arrays and B = A007318. - Philippe Deléham, Dec 08 2009
T(n,k) = T(n-1,k-1) + Sum_{i>=0} T(n-1,k+1+i)*3^i, T(0,0) = 1. - Philippe Deléham, Feb 23 2012
T(n,k) = ((k+1)/(n+1))*Sum_{j=0..floor((n-k)/2)} 3^(n-k-2*j)*C(n+1,j)*C(n-k-j-1,n-k-2*j). - Vladimir Kruchinin, Apr 04 2019

Terms a(55) onward added by G. C. Greubel, Apr 04 2019

A364647 G.f. satisfies A(x) = 1/(1 - 5x) - xA(x)^3.

Original entry on oeis.org

1, 4, 13, 38, 135, 677, 3538, 15868, 63313, 268430, 1348190, 7038185, 33328258, 144159428, 642323050, 3213846836, 16700677289, 80935833050, 363843867265, 1660048399600, 8276473557820, 42830085070355, 210286731046320, 967456811687945, 4476690297795850

Offset: 0

Seiichi Manyama, Jul 31 2023

nonn

Cf. A364641, A364645, A364646.

Cf. A026378.

a(n) = sum(k=0, n, (-1)^k*5^(n-k)*binomial(n+k, 2*k)*binomial(3*k, k)/(2*k+1));

a(n) = Sum_{k=0..n} (-1)^k * 5^(n-k) * binomial(n+k,2*k) * binomial(3*k,k) / (2*k+1).

A026380 a(n) = T(n,[ n/2 ]), where T is the array in A026374.

Original entry on oeis.org

1, 3, 4, 11, 17, 45, 75, 195, 339, 873, 1558, 3989, 7247, 18483, 34016, 86515, 160795, 408105, 764388, 1936881, 3650571, 9238023, 17501619, 44241261, 84179877, 212601015, 406020930, 1024642875, 1963073865, 4950790605

Offset: 0

Clark Kimberling

nonn

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

Cf. A026375, A026378.

a(2n)=A026378(n+1), a(2n-1)=A026375(n). - Emeric Deutsch, Feb 18 2004
a(2n) = A026378(2n+1), a(2n+1) = A026375(n+1).
Davenport et al. give a g.f.

A084536 Triangular array related to Motzkin triangle A026300.

Original entry on oeis.org

1, 1, 3, 1, 6, 10, 1, 9, 29, 36, 1, 12, 57, 132, 137, 1, 15, 94, 315, 590, 543, 1, 18, 140, 612, 1629, 2628, 2219, 1, 21, 195, 1050, 3605, 8127, 11732, 9285, 1, 24, 259, 1656, 6950, 19992, 39718, 52608, 39587, 1, 27, 332, 2457, 12177, 42498, 106644, 191754

Offset: 0

N. J. A. Sloane, Jun 29 2003

Reversal of A091965. Row sums give A026378. - Philippe Deléham, Mar 23 2007

			Triangle begins
  1;
  1,   3;
  1,   6,  10;
  1,   9,  29,  36;
  1,  12,  57, 132, 137; ...

A. Nkwanta, Lattice paths and RNA secondary structures, in African Americans in Mathematics, ed. N. Dean, Amer. Math. Soc., 1997, pp. 137-147.

T(n,n) = A002212(n+1). - Philippe Deléham, Mar 23 2007

More terms from Philippe Deléham, Mar 23 2007

A124431 a(n) = Sum_{k=0..n} 2^kC([(n+k)/2],k)C([(n+k+1)/2],k) where [x]=floor(x).

Original entry on oeis.org

1, 3, 9, 29, 97, 331, 1145, 4001, 14089, 49915, 177713, 635293, 2278841, 8198227, 29567729, 106872961, 387038993, 1404052659, 5101219929, 18559193245, 67605310097, 246541193883, 899999057385, 3288522934433, 12026324883865

Offset: 0

Paul D. Hanna, Oct 31 2006

nonn

This is the inverse Motzkin transform of A026378 assuming offset 1 here. - R. J. Mathar, Jul 07 2009
Hankel transform is Somos-4 variant A162547. - Paul Barry, Jan 09 2011
a(n) is the number of peakless Motzkin paths of length n in which the (1,0)-steps at level 0 come in 3 colors and those at a higher level come in 2 colors. Example: a(3)=29 because, denoting  U=(1,1), H=(1,0), and D=(1,-1), we have 3^3 = 27 paths of shape HHH and 2 paths of shape UHD. - Emeric Deutsch, May 03 2011
Conjecture: (n+1)*a(n) -2*(2*n+1)*a(n-1) +2*(n-1)*a(n-2) +2*(5-2*n)*a(n-3) +(n-3)*a(n-4) =0. - R. J. Mathar, Aug 09 2012

			G.f. = 1 + 3*x + 9*x^2 + 29*x^3 + 97*x^4 + 331*x^5 + 1145*x^6 + 4001*x^7 + ...

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

Cf. A124428.

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (Sqrt((1+x^2)/(1-4*x+x^2)) -1)/(2*x) )); // G. C. Greubel, Feb 26 2019

Table[Sum[2^k Binomial[Floor[(n+k)/2],k]Binomial[Floor[(n+k+1)/2],k],{k,0,n}],{n,0,30}] (* Harvey P. Dale, May 20 2012 *)
CoefficientList[Series[(Sqrt[(1+x^2)/(1-4*x+x^2)] -1)/(2*x), {x,0,30}],x] (* G. C. Greubel, Feb 26 2019 *)

a(n)=sum(k=0,n,2^k*binomial((n+k)\2,k)*binomial((n+k+1)\2,k))

my(x='x+O('x^30)); Vec((sqrt((1+x^2)/(1-4*x+x^2)) -1)/(2*x)) \\ G. C. Greubel, Feb 26 2019

((sqrt((1+x^2)/(1-4*x+x^2)) -1)/(2*x)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Feb 26 2019

a(n) = Sum_{k=0..n} 2^k*A124428(n+k,k).
G.f.: (((x^2+1)*(1-4*x+x^2))^(1/2) - (1-4*x+x^2))/(2*x*(1-4*x+x^2)). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 12 2009
G.f.: (1/(1-4*x+x^2))*c(-x/(1-4*x+x^2)), c(x) the g.f. of A000108. - Paul Barry, Jan 09 2011
G.f.: G(0)/(2*x) - 1/(2*x), where G(k)= 1 + 4*x*(4*k+1)/( (1+x^2)*(4*k+2) - x*(1+x^2)*(4*k+2)*(4*k+3)/(x*(4*k+3) + (1+x^2)*(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 26 2013
a(n) ~ sqrt(14*sqrt(3)-24) * (2+sqrt(3))^(n+2) / (2*sqrt(3*Pi*n)). - Vaclav Kotesovec, Feb 03 2014
0 = a(n)*(+a(n+1) - 6*a(n+2) + 6*a(n+3) - 18*a(n+4) + 5*a(n+5)) + a(n+1)*(-2*a(n+1) + 14*a(n+2) - 10*a(n+3) + 61*a(n+4) - 18*a(n+5)) + a(n+2)*(+4*a(n+2) - 28*a(n+3) - 10*a(n+4) + 6*a(n+5)) + a(n+3)*(+4*a(n+3) + 14*a(n+4) - 6*a(n+5)) + a(n+4)*(-2*a(n+4) + a(n+5)) if n>=0. - Michael Somos, Aug 06 2014
Conjecture: +(n+1)*a(n) +2*(-2*n-1)*a(n-1) +2*(n-1)*a(n-2) +2*(-2*n+5)*a(n-3) +(n-3)*a(n-4)=0. - R. J. Mathar, Jun 17 2016

cp's OEIS Frontend

A026392 T(n,[ n/2 ]), where T is the array in A026386.

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A126182 Let P be Pascal's triangle A007318 and let N be Narayana's triangle A001263, both regarded as lower triangular matrices. Sequence gives triangle obtained from P*N, read by rows.

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PARI

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A171651 Triangle T, read by rows : T(n,k) = A007318(n,k)*A005773(n+1-k).

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A026387 a(n) = number of integer strings s(0),...,s(n) counted by array T in A026386 that have s(n)=0; also a(n) = T(2n,n).

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A129164 Sum of pyramid weights in all skew Dyck paths of semilength n.

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Maxima

Maxima

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A171224 Riordan array (f(x),x*f(x)) where f(x) is the g.f. of A117641.

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Magma

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Maxima

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A364647 G.f. satisfies A(x) = 1/(1 - 5*x) - x*A(x)^3.

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A364647 G.f. satisfies A(x) = 1/(1 - 5x) - xA(x)^3.

A124431 a(n) = Sum_{k=0..n} 2^kC([(n+k)/2],k)C([(n+k+1)/2],k) where [x]=floor(x).