A025565 -id:A025565 - cp's OEIS frontend

A239101 Riordan array read by rows, corresponding to array in A180562.

1, 2, 1, 4, 2, 1, 10, 5, 2, 1, 26, 13, 6, 2, 1, 70, 35, 16, 7, 2, 1, 192, 96, 45, 19, 8, 2, 1, 534, 267, 126, 56, 22, 9, 2, 1, 1500, 750, 357, 160, 68, 25, 10, 2, 1, 4246, 2123, 1016, 463, 198, 81, 28, 11, 2, 1, 12092, 6046, 2907, 1337, 586, 240, 95, 31

Offset: 0

N. J. A. Sloane, Mar 25 2014

Take lower triangle of square array in A180562, read from right to left.
Row sums are in A225034. - Philippe Deléham, Mar 25 2014
Riordan array (f(x), (f(x)-1)/(2*f(x))) where f(x) = sqrt((1+x)/(1-3*x)). - Philippe Deléham, Mar 25 2014

			Triangle begins:
1
2 1
4 2 1
10 5 2 1
26 13 6 2 1
70 35 16 7 2 1
192 96 45 19 8 2 1
...
192 = 2*96, 96 = 70 - 35 + 16 + 45, 45 = 35 - 16 + 7 + 19, etc. - _Philippe Deléham_, Mar 25 2014
Production matrix is:
2, 1
0, 0, 1
2, 1, 0, 1
2, 1, 1, 0, 1
2, 1, 1, 1, 0, 1
2, 1, 1, 1, 1, 0, 1
2, 1, 1, 1, 1, 1, 0, 1
2, 1, 1, 1, 1, 1, 1, 0, 1
... _Philippe Deléham_, Sep 15 2014

D. Baccherini, D. Merlini, R. Sprugnoli, Binary words excluding a pattern and proper Riordan arrays, Discrete Math. 307 (2007), no. 9-10, 1021--1037. MR2292531 (2008a:05003).

Cf. A180562.

Cf. T(n,0) = A025565(n+1), T(n+1,1) = A005773(n+1), T(n+2,2) = A005717(n+1), A225034 (Row sums). - Philippe Deléham, Mar 25 2014

T(0,0) = 1, T(n,0) = 2*T(n,1) for n>0, T(n,k) = T(n-1,k-1) - T(n-1,k) + T(n-1,k+1) + T(n,k+1) for k>0, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Mar 25 2014

More terms from Philippe Deléham, Mar 25 2014

A079213 Triangle read by rows giving T(n,k) = number of sets of k black squares in an n X n checkerboard with the upper left corner colored black, such that the line joining any 2 squares slopes down to the right, 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 5, 3, 1, 1, 8, 12, 4, 1, 1, 13, 28, 22, 5, 1, 1, 18, 63, 68, 35, 6, 1, 1, 25, 117, 197, 135, 51, 7, 1, 1, 32, 208, 464, 480, 236, 70, 8, 1, 1, 41, 336, 1016, 1376, 996, 378, 92, 9, 1, 1, 50, 525, 2000, 3600, 3372, 1848, 568, 117, 10, 1, 1, 61, 775, 3725

Offset: 0

Dean Hickerson, Jan 02 2003

Based on a question from Cees H. Elzinga (ch.elzinga(AT)tiscali.nl), Dec 30 2002

			T(5,3)=22; one of the 22 sets of 3 is shown by the asterisks below; the 'o's denote black squares not in the set.
*.o.o
.*.o.
o.o.*
.o.o.
o.o.o

Cf. A025565.

f[m_, n_, 0] := 1; f[m_, n_, k_] := f[m, n, k]=Sum[If[EvenQ[m+n+mp+np], f[mp, np, k-1], 0], {mp, k-1, m-1}, {np, k-1, n-1}]; T[n_, k_] := f[n, n, k]; Flatten[Table[T[n, k], {n, 0, 11}, {k, 0, n}]]

More generally, let f(m, n, k) be the number of such sets in an m X n checkerboard. Then f(m, n, k) = Sum_{k-1<=m'

G.f.: Sum_{m>=0, n>=0, k>=0} f(m, n, k) x^m * y^n * z^k = (1+x) * (1+y) / ((1-x^2) * (1-y^2) + x*y*z*(1+x*y)).

T(n, 0) = T(n, n) = 1. T(n, 1) = ceiling(n^2/2). T(n, 2) = (n^2 * (n^2-2*n+4))/16 if n is even, ((n-1)^2 * (n^2+3))/16 if n is odd. T(n, n-1) = n. T(n, n-2) = (n-1)*(3n-4)/2.

G.f. (conjectured): Sum_{n>=0, k>=0} T(n, k) x^n y^k = sqrt((1+x)/((1+x-x*y)((1-x)^2 - x*y*(1+x)))).

Conjecture: Sum_{k=0..n} T(n, k) = A025565(n+1).

A352916 a(n) = A025179(n-2) + A102839(n-4), for n >= 4, with a(0) = a(2) = 0 and a(1) = a(3) = 1.

Original entry on oeis.org

0, 1, 0, 1, 1, 5, 13, 41, 121, 366, 1100, 3319, 10015, 30253, 91433, 276475, 836291, 2530321, 7657317, 23175867, 70150875, 212349687, 642803631, 1945819299, 5890003539, 17828324220, 53961228258, 163314594513, 494238394601, 1495593167851, 4525366817455

Offset: 0

Kassie Archer, Apr 26 2022

nonn

a(n) + 2*A025565(n) is the number of Dyck paths of semilength n+2 with L(D) = 4 where L(D) is the product of binomial coefficients (u_i(D)+d_i(D) choose u_i(D)), where u_i(D) is the number of up-steps between the i-th and (i+1)-st down step and d_i(D) is the number of down-steps between the i-th and (i+1)-st up step.

Kassie Archer and Christina Graves, Pattern-restricted permutations composed of 3-cycles, arXiv:2104.12664 [math.CO], 2021.
Kassie Archer and Christina Graves, A new statistic on Dyck paths for counting 3-dimensional Catalan words, arXiv:2205.09686 [math.CO], 2022.

Cf. A001006, A025179, A025565, A102839.

a:= proc(n) option remember; `if`(n<4, irem(n, 2),
     ((3*(n-4))*(n^4+6*n^3-41*n^2+18*n+76)*a(n-2)+
      (2*n^5+3*n^4-136*n^3+525*n^2-658*n+132)*a(n-1))/
      ((n^4+2*n^3-53*n^2+114*n+12)*(n-1)))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, May 19 2022

m[n_] := m[n] = If[n == 0, 1, m[n-1] + Sum[m[k]*m[n-2-k], {k, 0, n-2}]];
a[n_] := Switch[n, 0|2, 0, 1|3|4, 1, _, m[n-3] + Binomial[n-3, 2]*m[n-5] + 2*Sum[(i+1)*m[i]*m[n-5-i], {i, 0, n-5}]];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Dec 26 2022 *)

m(n) = polcoeff( ( 1 - x - sqrt((1 - x)^2 - 4 * x^2 + x^3 * O(x^n))) / (2 * x^2), n); \\ A001006
a(n) = if (n<=3, n%2, m(n-3) + binomial(n-3,2)*m(n-5) + 2*sum(i=0, n-5, (i+1)*m(i)*m(n-5-i))); \\ Michel Marcus, May 19 2022

a(n) = m(n-3) + binomial(n-3,2)*m(n-5) + 2*Sum_{i=0..n-5} (i+1)*m(i)*m(n-5-i) for n>=3, where m(n) = A001006(n) is the n-th Motzkin number.
a(n) ~ sqrt(n) * 3^(n - 7/2) / (4*sqrt(Pi)). - Vaclav Kotesovec, Jun 03 2022
D-finite with recurrence (n-1)*(n-69)*a(n) +(n^2+221*n-513)*a(n-1) +(-37*n^2+356*n-312)*a(n-2) +(47*n^2-859*n+2820)*a(n-3) +12*(7*n-30)*(n-6)*a(n-4)=0. - R. J. Mathar, Jul 17 2023

A353133 Coefficients of expansion of f(x) = (1+xm(x))^5(x^2(xm(x))'+1) where m(x) is the generating function for A001006.

Original entry on oeis.org

1, 5, 16, 47, 136, 392, 1130, 3262, 9434, 27337, 79364, 230815, 672380, 1961635, 5730860, 16763685, 49093260, 143924943, 422352816, 1240529133, 3646710456, 10728322770, 31584554610, 93048320820, 274292367650, 809044988695, 2387642856380, 7050001551361, 20826624824612, 61552574382856

Offset: 0

Kassie Archer, Apr 25 2022

nonn

2*x^7*f(x) is the generating function for the number of Dyck paths with L(D)=7 where L(D) is the product of binomial coefficients (u_i(D)+d_i(D) choose u_i(D)), where u_i(D) is the number of up-steps between the i-th and (i+1)-st down step and d_i(D) is the number of down-steps between the i-th and (i+1)-st up step.

Kassie Archer and Christina Graves, Pattern-restricted permutations composed of 3-cycles, arXiv:2104.12664 [math.CO], 2021.
Kassie Archer and Christina Graves, A new statistic on Dyck paths for counting 3-dimensional Catalan words, arXiv:2205.09686 [math.CO], 2022.

Cf. A001006, A005717, A005773, A025565, A225034.

m(x) = (1-x-sqrt(1-2*x-3*(x^2)))/(2*(x^2));
my(x='x+O('x^30)); Vec((1+x*m(x))^5*(x^2*(x*m(x))'+1)) \\ Michel Marcus, Apr 25 2022

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A239101 Riordan array read by rows, corresponding to array in A180562.

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A079213 Triangle read by rows giving T(n,k) = number of sets of k black squares in an n X n checkerboard with the upper left corner colored black, such that the line joining any 2 squares slopes down to the right, 0 <= k <= n.

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A352916 a(n) = A025179(n-2) + A102839(n-4), for n >= 4, with a(0) = a(2) = 0 and a(1) = a(3) = 1.

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A353133 Coefficients of expansion of f(x) = (1+x*m(x))^5*(x^2*(x*m(x))'+1) where m(x) is the generating function for A001006.

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A353133 Coefficients of expansion of f(x) = (1+xm(x))^5(x^2(xm(x))'+1) where m(x) is the generating function for A001006.