A051286 -id:A051286 - cp's OEIS frontend

A299499 Triangle read by rows, T(n,k) = [x^k] Sum_{k=0..n} p_{n,k}(x) where p_{n,k}(x) = x^kbinomial(n, k)hypergeom([-k, k-n, k-n], [1, -n], 1/x), for 0 <= k <= n.

1, 1, 1, 2, 2, 1, 5, 5, 3, 1, 11, 16, 9, 4, 1, 26, 44, 34, 14, 5, 1, 63, 122, 111, 60, 20, 6, 1, 153, 341, 351, 225, 95, 27, 7, 1, 376, 940, 1103, 796, 400, 140, 35, 8, 1, 931, 2581, 3384, 2764, 1561, 651, 196, 44, 9, 1, 2317, 7064, 10224, 9304, 5915, 2772, 994, 264, 54, 10, 1

Offset: 0

Peter Luschny, Feb 11 2018

			The partial polynomials p_{n,k}(x) start:
[0] 1
[1] 1, x
[2] 1, 2*x+ 1,    x^2
[3] 1, 3*x+ 4,  3*x^2+ 2*x,      x^3
[4] 1, 4*x+ 9,  6*x^2+12*x+1,  4*x^3+ 3*x^2,       x^4
[5] 1, 5*x+16, 10*x^2+36*x+9, 10*x^3+24*x^2+3*x, 5*x^4+4*x^3, x^5
.
The polynomials P_{n}(x) start:
[0]   1
[1]   1 +    x
[2]   2 +  2*x +    x^2
[3]   5 +  5*x +  3*x^2 +    x^3
[4]  11 + 16*x +  9*x^2 +  4*x^3 +   x^4
[5]  26 + 44*x + 34*x^2 + 14*x^3 + 5*x^4 + x^5
.
The triangle starts:
[0]   1
[1]   1,    1
[2]   2,    2,    1
[3]   5,    5,    3,    1
[4]  11,   16,    9,    4,    1
[5]  26,   44,   34,   14,    5,   1
[6]  63,  122,  111,   60,   20,   6,   1
[7] 153,  341,  351,  225,   95,  27,   7,  1
[8] 376,  940, 1103,  796,  400, 140,  35,  8, 1
[9] 931, 2581, 3384, 2764, 1561, 651, 196, 44, 9, 1
.
The square array P_{n}(k) near k=0:
......  [k=-2] 1, -1,  2, -1,  -1,   10,  -25,    51,    -68,     41, ...
A182883 [k=-1] 1,  0,  1,  2,   1,    6,    7,    12,     31,     40, ...
A051286 [k=0]  1,  1,  2,  5,  11,   26,   63,   153,    376,    931, ...
A108626 [k=1]  1,  2,  5, 14,  41,  124,  383,  1200,   3799,  12122, ...
A299443 [k=2]  1,  3, 10, 35, 127,  474, 1807,  6999,  27436, 108541, ...
......  [k=3]  1,  4, 17, 74, 329, 1490, 6855, 31956, 150607, 716236, ...

Cf. A051286, A108625, A108626, A182883, A298611, A299443, A299500.

CoeffList := p -> op(PolynomialTools:-CoefficientList(p,x)):
PrintPoly := p -> print(sort(expand(p),x,ascending)):
T := (n,k) -> x^k*binomial(n, k)*hypergeom([-k,k-n,k-n], [1,-n], 1/x):
P := [seq(add(simplify(T(n,k)),k=0..n), n=0..10)]:
seq(CoeffList(p), p in P); seq(PrintPoly(p), p in P);
R := proc(n,k) option remember; # Recurrence
if n < 4 then return [1,k+1,(k+1)^2+1,(k+1)^3+2*k+4][n+1] fi; ((2-n)*R(n-4,k)+
(3-2*n)*(k-1)*R(n-3,k)+(k^2+2*k-1)*(1-n)*R(n-2,k)+(2*n-1)*(k+1)*R(n-1,k))/n end:
for k from -2 to 3 do lprint(seq(R(n,k), n=0..9)) od;

nmax = 10;
p[n_, k_, x_] := x^k*Binomial[n, k]*HypergeometricPFQ[{-k, k-n, k-n}, {1, -n}, 1/x];
p[n_, x_] := Sum[p[n, k, x], {k, 0, n}];
Table[CoefficientList[p[n, x], x], {n, 0, nmax}] // Flatten (* Jean-François Alcover, Feb 26 2018 *)

Let P_{n}(x) = Sum_{k=0..n} p_{n,k}(x) then
2^n*P_{n}(1/2) = A298611(n).
P_{n}(-1) = A182883(n), P_{n}(0) = A051286(n).
P_{n}( 1) = A108626(n), P_{n}(2) = A299443(n).
The general case: for fixed k the sequence P_{n}(k) with n >= 0 has the generating function ogf(k, x) = (1-2*(k+1)*x + (k^2+2*k-1)*x^2 + 2*(k-1)*x^3 + x^4)^(-1/2).  The example section shows the start of this square array of sequences.
These sequences can be computed by the recurrence P(n,k) = ((2-n)*P(n-4,k)+(3-2*n)*(k-1)*P(n-3,k)+(k^2+2*k-1)*(1-n)*P(n-2,k)+(2*n-1)*(k+1)*P(n-1,k))/n with initial values 1, k+1, (k+1)^2+1 and (k+1)^3+2*k+4.
The partial polynomials p_{n,k}(x) reduce for x = 1 to A108625 (seen as a triangle).

A323767 A(n,k) = Sum_{j=0..floor(n/2)} binomial(n-j,j)^k, square array A(n,k) read by antidiagonals, for n >= 0, k >= 0.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 3, 3, 1, 1, 2, 5, 5, 3, 1, 1, 2, 9, 11, 8, 4, 1, 1, 2, 17, 29, 26, 13, 4, 1, 1, 2, 33, 83, 92, 63, 21, 5, 1, 1, 2, 65, 245, 338, 343, 153, 34, 5, 1, 1, 2, 129, 731, 1268, 1923, 1281, 376, 55, 6

Offset: 0

Seiichi Manyama, Jan 27 2019

			Square array begins:
   1,  1,   1,    1,     1,      1,       1, ...
   1,  1,   1,    1,     1,      1,       1, ...
   2,  2,   2,    2,     2,      2,       2, ...
   2,  3,   5,    9,    17,     33,      65, ...
   3,  5,  11,   29,    83,    245,     731, ...
   3,  8,  26,   92,   338,   1268,    4826, ...
   4, 13,  63,  343,  1923,  10903,   62283, ...
   4, 21, 153, 1281, 11553, 108801, 1050753, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns 0-5 give A004526(n+2), A000045(n+1), A051286, A181545, A181546, A181547.
Rows 0-5 give A000012, A000012, A007395, A000051, A168607, A074506.
Main diagonal gives A323769.
Cf. A011973,

f := Sum[Power[Binomial[#1 - i, i], #2], {i, 0, #1/2}] &;a = Flatten[Reverse[DeleteCases[Table[Table[f[m - n, n], {n, 0, 20}], {m, 0, 20}], 0, Infinity], 2]] (* Elijah Beregovsky, Nov 24 2020 *)

A382841 a(n) = Sum_{k=0..floor(n/2)} (binomial(n,k) * binomial(n-k,k))^2.

Original entry on oeis.org

1, 1, 5, 37, 181, 1301, 9401, 65465, 498037, 3796021, 29221705, 230396585, 1828448425, 14651160265, 118544522045, 965075143037, 7907605360757, 65162569952245, 539515760866889, 4486877961224297, 37463151704756281, 313909383754331801, 2638892573249746445, 22249830926517611917

Offset: 0

Ilya Gutkovskiy, Apr 06 2025

nonn

Diagonal of the rational function 1 / ((1 - x)*(1 - y)*(1 - z)*(1 - w) - (x*y)^2*z*w).

Alois P. Heinz, Table of n, a(n) for n = 0..1053

Cf. A000984, A002426, A005259, A005260, A051286, A181546, A275027, A277247, A382842.

Cf. A089627.

a:= n-> add(combinat[multinomial](n, n-2*k, k$2)^2, k=0..n/2):
seq(a(n), n=0..23);  # Alois P. Heinz, Apr 07 2025

Table[Sum[(Binomial[n, k] Binomial[n - k, k])^2, {k, 0, Floor[n/2]}], {n, 0, 23}]
Table[HypergeometricPFQ[{1/2 - n/2, 1/2 - n/2, -n/2, -n/2}, {1, 1, 1}, 16], {n, 0, 23}]
Table[SeriesCoefficient[1/((1 - x) (1 - y) (1 - z) (1 - w) - (x y)^2 z w), {x, 0, n}, {y, 0, n}, {z, 0, n}, {w, 0, n}], {n, 0, 23}]

a(n) ~ 3^(2*n+2) / (2^(5/2) * Pi^(3/2) * n^(3/2)). - Vaclav Kotesovec, Apr 07 2025

a(n) = Sum_{k=0..floor(n/2)} A089627(n,k)^2. - Alois P. Heinz, Apr 07 2025

A125250 Square array, read by antidiagonals, where A(1,1) = A(2,2) = 1, A(1,2) = A(2,1) = 0, A(n,k) = 0 if n < 1 or k < 1, otherwise A(n,k) = A(n-2,k-2) + A(n-1,k-2) + A(n-2,k-1) + A(n-1,k-1).

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 2, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 1, 5, 1, 0, 0, 0, 0, 0, 5, 5, 0, 0, 0, 0, 0, 0, 3, 11, 3, 0, 0, 0, 0, 0, 0, 1, 13, 13, 1, 0, 0, 0, 0, 0, 0, 0, 9, 26, 9, 0, 0, 0, 0, 0, 0, 0, 0, 4, 32, 32, 4, 0, 0, 0, 0, 0, 0, 0, 0, 1, 26, 63, 26, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 14, 80, 80, 14, 0, 0, 0, 0, 0

Offset: 1

Gerald McGarvey, Jan 15 2007

It appears that the main diagonal (1,1,2,5,11,...) is A051286 (Whitney number of level n of the lattice of the ideals of the fence of size 2 n) that the diagonals (0,1,2,5,13,...) adjacent to the main diagonal are A110320 (Number of blocks in all RNA secondary structures with n nodes) and that the n-th antidiagonal sum = A094686(n-1) (a Fibonacci convolution). The n-th row sum = A002605(n).

			Array starts as:
1 0 0 0  0  0  0 ...
0 1 1 0  0  0  0 ...
0 1 2 2  1  0  0 ...
0 0 2 5  5  3  1   0 ...
0 0 1 5 11 13  9   4   1   0...
0 0 0 3 13 26 32  26  14   5   1  0 ...
0 0 0 1  9 32 63  80  71  45  20  6  1 0 ...
0 0 0 0  4 26 80 153 201 191 135 71 27 7 1 0 ...
...

Cf. A051286, A110320, A002605, A026729, A030528, A057089, A109466.

T[n_, k_] := Sum[Binomial[i, n-i] Binomial[i, k-i], {i, Floor[(n+1)/2], k}];
Table[T[n-k, k], {n, 0, 13}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 12 2019 *)

A=matrix(22,22);A[1,1]=1;A[2,2]=1;A[2,1]=0;A[1,2]=0;A[3,2]=1;A[2,3]=1; for(n=3,22,for(k=3,22,A[n,k]=A[n-2,k-2]+A[n-1,k-2]+A[n-2,k-1]+A[n-1,k-1])); for(n=1,22,for(i=1,n,print1(A[n-i+1,i],", ")))

A(1,1) = A(2,2) = 1, A(1,2) = A(2,1) = 0, A(n,k) = 0 if n < 1 or k < 1, otherwise A(n,k) = A(n-2,k-2) + A(n-1,k-2) + A(n-2,k-1) + A(n-1,k-1).
From Peter Bala, Nov 07 2017: (Start)
T(n,k) = Sum_{i = floor((n+1)/2)..k} binomial(i,n-i)* binomial(i,k-i).
Square array = A026729 * transpose(A026729), where A026729 is viewed as a lower unit triangular array. Omitting the first row and column of square array = A030528 * transpose(A030528).
O.g.f. 1/(1 - t*(1 + t)*x - t*(1 + t)*x^2) = 1 + (t + t^2)*x + (t + 2*t^2 + 2*t^3 + t^4)*x^2 + .... Cf. A109466 with o.g.f. 1/(1 - t*x - t*x^2).
The n-th row polynomial R(n,t) satisfies R(n,t) = R(n,-1 - t).
R(n,t) = (-1)^n*sqrt(-t*(1 + t))^n*U(n, 1/2*sqrt(-t*(1 + t))), where U(n,x) denotes the n-th Chebyshev polynomial of the second kind.
The sequence of row polynomials R(n,t) is a divisibility sequence of polynomials, that is, if m divides n then R(m,t) divides R(n,t) in the polynomial ring Z[t].
R(n,1) = A002605; R(n,2) = A057089. (End)

A182880 Triangle read by rows: T(n,k) is the number of weighted lattice paths in L_n having k (1,1)-steps. L_n is the set of lattice paths of weight n that start at (0,0) and end on the horizontal axis and whose steps are of the following four kinds: a (1,0)-step with weight 1; a (1,0)-step with weight 2; a (1,1)-step with weight 2; a (1,-1)-step with weight 1. The weight of a path is the sum of the weights of its steps.

Original entry on oeis.org

1, 1, 2, 3, 2, 5, 6, 8, 18, 13, 44, 6, 21, 102, 30, 34, 222, 120, 55, 466, 390, 20, 89, 948, 1140, 140, 144, 1884, 3066, 700, 233, 3672, 7770, 2800, 70, 377, 7044, 18780, 9800, 630, 610, 13332, 43710, 31080, 3780, 987, 24946, 98610, 91560, 17850, 252, 1597, 46218, 216732, 254400, 72450, 2772

Offset: 0

Emeric Deutsch, Dec 11 2010

Sum of entries in row n is A051286(n).

			T(3,1)=2. Indeed, denoting by h (H) the (1,0)-step of weight 1 (2), and u=(1,1), d=(1,-1), the five paths of weight 3 are ud, du, hH, Hh, and hhh; two of them, ud and du, have exactly one u step.
Triangle starts:
   1;
   1;
   2;
   3,  2;
   5,  6;
   8, 18;
  13, 44, 6;

M. Bona and A. Knopfmacher, On the probability that certain compositions have the same number of parts, Ann. Comb., 14 (2010), 291-306.
E. Munarini, N. Zagaglia Salvi, On the rank polynomial of the lattice of order ideals of fences and crowns, Discrete Mathematics 259 (2002), 163-177.

Cf. A051286, A000045, A182881.

G:=1/sqrt(1-2*z-z^2+2*z^3+z^4-4*t*z^3): Gser:=simplify(series(G,z=0,18)): for n from 0 to 16 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 0 to 16 do seq(coeff(P[n],t,k),k=0..floor(n/3)) od; # yields sequence in triangular form

T(n,0) = A000045(n+1) (the Fibonacci numbers).
Sum_{k=0..n} k*T(n,k) = A182881(n).
G.f.: G(t,z) = 1/sqrt(1 - 2*z - z^2 + 2*z^3 + z^4 - 4*t*z^3).
The g.f. of column k is binomial(2n,n)*z^(3n)/(1-z-z^2)^(2n+1).
Apparently, T(n,1) = 2*A001628(n-3), T(n,2) = 6*A001873(n-6), T(n,3) = 20*A001875(n-9). - R. J. Mathar, Dec 11 2010

A182891 Triangle read by rows: T(n,k) is the number of weighted lattice paths in L_n having k (1,0)-steps of weight 2 at level 0. The members of L_n are paths of weight n that start at (0,0) , end on the horizontal axis and whose steps are of the following four kinds: an (1,0)-step with weight 1, an (1,0)-step with weight 2, a (1,1)-step with weight 2, and a (1,-1)-step with weight 1. The weight of a path is the sum of the weights of its steps.

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 7, 3, 1, 15, 8, 3, 35, 21, 6, 1, 83, 50, 16, 4, 197, 123, 45, 10, 1, 473, 308, 117, 28, 5, 1145, 769, 304, 83, 15, 1, 2787, 1926, 798, 232, 45, 6, 6819, 4843, 2085, 636, 140, 21, 1, 16759, 12204, 5433, 1744, 416, 68, 7, 41345, 30813, 14154, 4749, 1200, 222, 28, 1

Offset: 0

Emeric Deutsch, Dec 12 2010

Sum of entries in row n is A051286(n).
T(n,0)=A182892(n).
Sum(k*T(n,k), k=0..n)=A182890(n-1).

			T(3,1)=2. Indeed, denoting by h (H) the (1,0)-step of weight 1 (2), and u=(1,1), d=(1,-1), the five paths of weight 3 are ud, du, hH, Hh, and hhh; two of them, namely hH and Hh, have exactly one H-step at level 0.
Triangle starts:
1;
1;
1,1;
3,2;
7,3,1;
15,8,3;
35,21,6,1;

M. Bona and A. Knopfmacher, On the probability that certain compositions have the same number of parts, Ann. Comb., 14 (2010), 291-306.
E. Munarini, N. Zagaglia Salvi, On the rank polynomial of the lattice of order ideals of fences and crowns, Discrete Mathematics 259 (2002), 163-177.

Cf. A051286, A182890, A182892.

G:=1/(z^2-t*z^2+sqrt((1+z+z^2)*(1-3*z+z^2))): Gser:=simplify(series(G,z=0,18)): for n from 0 to 14 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 0 to 14 do seq(coeff(P[n],t,k),k=0..floor(n/2)) od; # yields sequence in triangular form

G.f. G(t,z) =1/[z^2-tz^2+sqrt((1+z+z^2)(1-3z+z^2))].

A182898 Triangle read by rows: T(n,k) is the number of weighted lattice paths in L_n having k returns to the horizontal axis (both from above and below). The members of L_n are paths of weight n that start at (0,0) , end on the horizontal axis and whose steps are of the following four kinds: an (1,0)-step with weight 1, an (1,0)-step with weight 2, a (1,1)-step with weight 2, and a (1,-1)-step with weight 1. The weight of a path is the sum of the weights of its steps.

Original entry on oeis.org

1, 1, 2, 3, 2, 5, 6, 8, 18, 13, 46, 4, 21, 112, 20, 34, 262, 80, 55, 600, 268, 8, 89, 1356, 816, 56, 144, 3046, 2324, 280, 233, 6832, 6320, 1144, 16, 377, 15354, 16620, 4136, 144, 610, 34658, 42652, 13728, 864, 987, 78706, 107520, 42816, 4144, 32, 1597, 180000, 267564, 127392, 17264, 352

Offset: 0

Emeric Deutsch, Dec 13 2010

Sum of entries in row n is A051286(n).
T(n,0)=A000045(n+1) (the Fibonacci numbers).
Sum(k*T(n,k), k=0..n)=A182899(n).

			T(3,1)=2. Indeed, denoting by h (H) the (1,0)-step of weight 1 (2), and u=(1,1), d=(1,-1), the five paths of weight 3 are ud, du, hH, Hh, and hhh; two of them, namely ud and du, have 1 return to the horizontal axis.
Triangle starts:
1;
1;
2;
3,2;
5,6;
8,18;
13,46,4;
21,112,20;

M. Bona and A. Knopfmacher, On the probability that certain compositions have the same number of parts, Ann. Comb., 14 (2010), 291-306.
E. Munarini, N. Zagaglia Salvi, On the rank polynomial of the lattice of order ideals of fences and crowns, Discrete Mathematics 259 (2002), 163-177.

eq := c = 1+z*c+z^2*c+z^3*c^2: c := RootOf(eq, c): G := 1/(1-z-z^2-2*t*z^3*c): Gser := simplify(series(G, z = 0, 20)): for n from 0 to 16 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 16 do seq(coeff(P[n], t, k), k = 0 .. floor((1/3)*n)) end do; # yields sequence in triangular form

G.f.: G(t,z) =1/[1-z-z^2-2tz^3*c], where c satisfies c = 1+zc+z^2*c+z^3*c^2.

The trivariate g.f. H=H(t,s,z), where t (s) marks (1,-1)-returns ((1,1)-returns) to the horizontal axis, and z marks weight is given by H=1+zH+z^2*H+(t+s)z^3*cH, where c satisfies c = 1+zc+z^2*c+z^3*c^2.

A182903 Triangle read by rows: T(n,k) is the number of weighted lattice paths in L_n having k peaks.The members of L_n are paths of weight n that start at (0,0), end on the horizontal axis and whose steps are of the following four kinds: an (1,0)-step with weight 1, an (1,0)-step with weight 2, a (1,1)-step with weight 2, and a (1,-1)-step with weight 1. The weight of a path is the sum of the weights of its steps. A peak is a (1,1)-step followed by a (1,-1)-step.

Original entry on oeis.org

1, 1, 2, 4, 1, 9, 2, 21, 5, 48, 14, 1, 112, 38, 3, 263, 104, 9, 623, 276, 31, 1, 1484, 730, 99, 4, 3550, 1921, 309, 14, 8525, 5034, 929, 56, 1, 20537, 13145, 2739, 205, 5, 49612, 34208, 7956, 716, 20, 120136, 88780, 22804, 2394, 90, 1, 291519, 229860, 64650

Offset: 0

Emeric Deutsch, Dec 16 2010

Number of entries in row n is 1+floor(n/3).
Sum of entries in row n is A051286(n).
T(n,0)= A182904(n).
Sum(k*T(n,k), k>=0)=A182884(n-2).

			T(7,2)=3. Indeed, denoting by h (H) the (1,0)-step of weight 1 (2), and U=(1,1), D=(1,-1), we have hUDUD, UDhUD, UDUDh.
Triangle starts:
1;
1;
2;
4,1;
9,2;
21,5;
48,14,1;

M. Bona and A. Knopfmacher, On the probability that certain compositions have the same number of parts, Ann. Comb., 14 (2010), 291-306.
E. Munarini, N. Zagaglia Salvi, On the rank polynomial of the lattice of order ideals of fences and crowns, Discrete Mathematics 259 (2002), 163-177.

Cf. A051286, A182904, A182884, A182900.

Let F=F(t,s,x,y,z) be the 5-variate g.f. of the considered weighted lattice paths, where z marks weight, t (s) marks number of peaks (valleys), x (y) indicates that the path starts with a (1,1)-step ((1,-1)-step). Then F(t,s,x,y,z)=1+z(1+z)F(t,s,1,1,z)+xz^3[t+H(t,s,z)-1]F(t,s,s,1,z)+yz^3[s+H(s,t,z)-1]F(t,s,1,t,z), where H=H(t,s,z) is given by H=1+zH+z^2*H+z^3*(t-1+H)[s(H-1-zH-z^2*H)+1+zH+z^2*H] (see A182900).

A183146 G.f.: Sum_{n>=0} [Sum_{k=0..n} C(n,k)^2x^k]^3 x^n.

Original entry on oeis.org

1, 1, 4, 16, 80, 407, 2221, 12380, 71196, 417016, 2484839, 15001779, 91603298, 564661194, 3509278042, 21964437947, 138330334357, 875977578584, 5574225259696, 35626247068500, 228592067446715, 1471959684881231

Offset: 0

Paul D. Hanna, Dec 26 2010

nonn

Compare g.f. to a g.f. of the Whitney numbers in A051286:

Sum_{n>=0} [Sum_{k=0..n} C(n,k)^2*x^k] * x^n.

			G.f.: A(x) = 1 + x + 4*x^2 + 16*x^3 + 80*x^4 + 407*x^5 + 2221*x^6 +...
which equals the sum of the series:
A(x) = 1 + (1 + x)^3*x + (1 + 4*x + x^2)^3*x^2
+ (1 + 9*x + 9*x^2 + x^3)^3*x^3
+ (1 + 16*x + 36*x^2 + 16*x^3 + x^4)^3*x^4
+ (1 + 25*x + 100*x^2 + 100*x^3 + 25*x^4 + x^5)^3*x^5
+ (1 + 36*x + 225*x^2 + 400*x^3 + 225*x^4 + 36*x^5 + x^6)^3*x^6 +...

Cf. A180717, A051286.

{a(n)=polcoeff(sum(m=0,n,sum(k=0,m,binomial(m,k)^2*x^k)^3*x^m)+x*O(x^n),n)}

A114711 Triangle read by rows: T(n,k) = number of peakless Motzkin paths of length n and having k weak ascents (1 <= k <= ceiling(n/3)).

Original entry on oeis.org

1, 1, 2, 3, 1, 5, 3, 8, 9, 13, 22, 2, 21, 51, 10, 34, 111, 40, 55, 233, 130, 5, 89, 474, 380, 35, 144, 942, 1022, 175, 233, 1836, 2590, 700, 14, 377, 3522, 6260, 2450, 126, 610, 6666, 14570, 7770, 756, 987, 12473, 32870, 22890, 3570, 42, 1597, 23109, 72244

Offset: 1

Emeric Deutsch, Dec 27 2005

A Motzkin path of length n is a lattice path from (0,0) to (n,0) consisting of U=(1,1), D=(1,-1) and H=(1,0) steps and never going below the x-axis. A weak ascent in a Motzkin path is a maximal sequence of consecutive U and H steps.

			T(5,2)=3 because we have (UH)D(UU), (UHH)D(H) and (HUH)D(H) (the weak ascents are shown between parentheses).
Triangle begins:
   1;
   1;
   2;
   3,  1;
   5,  3;
   8,  9;
  13, 22,  2;

Jean-Luc Baril and José Luis Ramírez, Descent distribution on Catalan words avoiding ordered pairs of Relations, arXiv:2302.12741 [math.CO], 2023.
I. L. Hofacker, P. Schuster and P. F. Stadler, Combinatorics of RNA secondary structures, Discrete Appl. Math., 88, 1998, 207-237.
P. R. Stein and M. S. Waterman, On some new sequences generalizing the Catalan and Motzkin numbers, Discrete Math., 26 (1978), 261-272.
M. Vauchassade de Chaumont and G. Viennot, Polynômes orthogonaux et problèmes d'énumération en biologie moléculaire, Publ. I.R.M.A. Strasbourg, 1984, 229/S-08, Actes 8e Sem. Lotharingien, pp. 79-86.

Cf. A004148, A000045, A001628, A000108, A051286.

G:=(1-z-z^2-sqrt(1-2*z-z^2+2*z^3+z^4-4*t*z^3))/2/z^2: Gser:=simplify(series(G,z=0,22)): for n from 1 to 18 do P[n]:=coeff(Gser,z^n) od: for n from 0 to 18 do seq(coeff(P[n],t^j),j=1..ceil(n/3)) od; # yields sequence in triangular form

Row n contains ceiling(n/3) terms.
Row sums yield the RNA secondary structure numbers (A004148).
Column 1 yields the Fibonacci numbers (A000045).
Column 2 yields A001628.
T(3n+1,n+1) = A000108(n) (the Catalan numbers).
Sum_{k=1..ceiling(n/3)} k*T(n,k) = A051286(n-1) (n >= 1).
G.f.: G = G(t, z) satisfies G = z*(t+G) + z^2*G*(1+G).

cp's OEIS Frontend

A299499 Triangle read by rows, T(n,k) = [x^k] Sum_{k=0..n} p_{n,k}(x) where p_{n,k}(x) = x^k*binomial(n, k)*hypergeom([-k, k-n, k-n], [1, -n], 1/x), for 0 <= k <= n.

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A323767 A(n,k) = Sum_{j=0..floor(n/2)} binomial(n-j,j)^k, square array A(n,k) read by antidiagonals, for n >= 0, k >= 0.

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A382841 a(n) = Sum_{k=0..floor(n/2)} (binomial(n,k) * binomial(n-k,k))^2.

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A125250 Square array, read by antidiagonals, where A(1,1) = A(2,2) = 1, A(1,2) = A(2,1) = 0, A(n,k) = 0 if n < 1 or k < 1, otherwise A(n,k) = A(n-2,k-2) + A(n-1,k-2) + A(n-2,k-1) + A(n-1,k-1).

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A183146 G.f.: Sum_{n>=0} [Sum_{k=0..n} C(n,k)^2*x^k]^3 * x^n.

A299499 Triangle read by rows, T(n,k) = [x^k] Sum_{k=0..n} p_{n,k}(x) where p_{n,k}(x) = x^kbinomial(n, k)hypergeom([-k, k-n, k-n], [1, -n], 1/x), for 0 <= k <= n.

A183146 G.f.: Sum_{n>=0} [Sum_{k=0..n} C(n,k)^2x^k]^3 x^n.