A369212
Expansion of (1/x) * Series_Reversion( x / ((1+x)^2+x^3) ).
Original entry on oeis.org
1, 2, 5, 15, 50, 177, 652, 2473, 9594, 37892, 151846, 615859, 2523217, 10427471, 43415259, 181941198, 766841846, 3248517320, 13823977350, 59067577266, 253315964424, 1089998388418, 4704475230340, 20361365646315, 88351705071583, 384280788724692, 1675063399090659
Offset: 0
-
my(N=30, x='x+O('x^N)); Vec(serreverse(x/((1+x)^2+x^3))/x)
-
a(n) = sum(k=0, n\3, binomial(n+1, k)*binomial(2*n-2*k+2, n-3*k))/(n+1);
A361932
G.f. A(x) satisfies A(x) = 1 + (x * A(x) / (1 - x))^3.
Original entry on oeis.org
1, 0, 0, 1, 3, 6, 13, 33, 84, 208, 522, 1341, 3476, 9042, 23673, 62426, 165504, 440664, 1178168, 3162357, 8517681, 23013294, 62356329, 169408107, 461366499, 1259311824, 3444497550, 9439766700, 25916832981, 71274793968, 196325540206, 541579442133
Offset: 0
-
a(n) = sum(k=0, n\3, binomial(n-1, n-3*k)*binomial(3*k, k)/(2*k+1));
A369481
Expansion of (1/x) * Series_Reversion( x / ((1+x) * (1+x+x^3)) ).
Original entry on oeis.org
1, 2, 5, 15, 51, 187, 718, 2844, 11530, 47612, 199576, 847013, 3632468, 15717041, 68527255, 300780438, 1327939406, 5893299392, 26275243626, 117635107818, 528631769323, 2383660351991, 10781500113896, 48903885040638, 222400899237943, 1013841791472632
Offset: 0
-
my(N=30, x='x+O('x^N)); Vec(serreverse(x/((1+x)*(1+x+x^3)))/x)
-
a(n, s=3, t=1, u=1) = sum(k=0, n\s, binomial(t*(n+1), k)*binomial((t+u)*(n+1)-k, n-s*k))/(n+1);
A369482
Expansion of (1/x) * Series_Reversion( x / ((1+x)^2 * (1+x+x^3)) ).
Original entry on oeis.org
1, 3, 12, 56, 287, 1564, 8895, 52195, 313655, 1920489, 11938271, 75143016, 477948051, 3067190311, 19835032603, 129129612163, 845603794947, 5566269982581, 36810651063798, 244448822313138, 1629413356387998, 10898124891668031, 73116947514706451
Offset: 0
-
my(N=30, x='x+O('x^N)); Vec(serreverse(x/((1+x)^2*(1+x+x^3)))/x)
-
a(n, s=3, t=1, u=2) = sum(k=0, n\s, binomial(t*(n+1), k)*binomial((t+u)*(n+1)-k, n-s*k))/(n+1);
A378849
a(n) is the total number of paths starting at (0,0), ending at (n,0), consisting of steps (1,1), (1,0), (1,-2), and staying on or above y = -1.
Original entry on oeis.org
1, 1, 1, 3, 9, 21, 48, 120, 309, 787, 2011, 5215, 13652, 35894, 94823, 251889, 672285, 1801185, 4842757, 13064059, 35349463, 95912989, 260896318, 711338596, 1943690464, 5321704006, 14597781706, 40112702176, 110404515703, 304338523999, 840140172621, 2322386563353
Offset: 0
-
a:= proc(n) option remember; `if`(n<4, [1$3, 3][n+1],
(2*(8*n^3+3*n^2-25*n-6)*a(n-1)-2*(n-1)*(12*n^2-9*n-10)*
a(n-2)+(43*n+13)*(n-1)*(n-2)*a(n-3)-31*(n-1)*(n-2)*
(n-3)*a(n-4))/(2*(2*n+3)*(n+3)*(n-2)))
end:
seq(a(n), n=0..31); # Alois P. Heinz, Dec 09 2024
-
a(n) = sum(k=0, floor(n/3), binomial(n, k*3)*binomial(3*k+1, k)/(k+1)) \\ Thomas Scheuerle, Dec 09 2024
A378850
a(n) is the total number of paths starting at (0, 0), ending at (n, 0), consisting of steps (1, 1), (1, 0), (1, -2), and staying on or above y = -2.
Original entry on oeis.org
1, 1, 1, 4, 13, 31, 73, 190, 505, 1316, 3431, 9065, 24122, 64325, 172082, 462344, 1246685, 3371135, 9140289, 24847422, 67708743, 184906614, 505986933, 1387240401, 3810083424, 10481797131, 28880894706, 79692785251, 220203155689, 609242057143, 1687666776031
Offset: 0
For n = 3, the a(3)=4 paths are DUU, HHH, UDU, UUD, where U=(1,1), D=(1,-2) and H=(1,0).
-
a := n -> hypergeom([4/3, 5/3, (1-n)/3, (2-n)/3, -n/3], [1/3, 2/3, 5/2, 2], -27/4):
seq(simplify(a(n)), n = 0..30); # Peter Luschny, Dec 18 2024
-
a(n) = sum(k=0, floor(n/3), binomial(n, k*3)*binomial(3*k+3, k+1)/(2*k+3)) \\ Thomas Scheuerle, Dec 09 2024
A127905
Construct triangle in which n-th row is obtained by expanding (1+x+x^3)^n and take the next-to-central column.
Original entry on oeis.org
0, 1, 2, 3, 8, 25, 66, 168, 456, 1269, 3490, 9581, 26544, 73944, 206220, 576045, 1613264, 4527661, 12725946, 35818135, 100950440, 284869263, 804726934, 2275500998, 6440230392, 18242735800, 51714552656
Offset: 0
-
[0] cat [n*(&+[Binomial(n-1,3*k)*Binomial(3*k,k)/(2*k+1): k in [0..Floor((n-1)/3)]]): n in [1..30]]; // G. C. Greubel, Apr 30 2018
-
A127905 := proc(n)
n*add(binomial(n-1,3*k)*binomial(3*k,k)/(2*k+1),k=0..floor((n-1)/3)) ;
end proc: # R. J. Mathar, Feb 23 2015
-
Table[n*Sum[Binomial[n-1,3*k]*Binomial[3*k,k]/(2*k+1), {k, 0, Floor[(n -1)/3]}], {n, 0, 50}] (* G. C. Greubel, Apr 30 2018 *)
-
a(n)=if(n<0, 0, polcoeff((1+x+x^3)^n, n-1));
-
a(n)=if(n<0, 0, n++; n*polcoeff(serreverse(x/(1+x+x^3)+x*O(x^n)), n))
A364539
G.f. satisfies A(x) = 1 + x*A(x) + x^3*A(x)^5.
Original entry on oeis.org
1, 1, 1, 2, 7, 22, 62, 182, 583, 1928, 6358, 21063, 70888, 241889, 831634, 2874584, 9995579, 34966279, 122938956, 434062141, 1538378816, 5471697241, 19525345791, 69880082323, 250767909528, 902123110483, 3252793321513, 11753570922933, 42553831219830
Offset: 0
-
a(n) = sum(k=0, n\3, binomial(n+2*k, 5*k)*binomial(5*k, k)/(4*k+1));
A365252
G.f. satisfies A(x) = 1 + x*A(x)*(1 + x^4*A(x)^2).
Original entry on oeis.org
1, 1, 1, 1, 1, 2, 5, 11, 21, 36, 60, 106, 205, 418, 851, 1685, 3257, 6264, 12210, 24276, 48920, 98873, 199118, 399472, 801361, 1613713, 3266772, 6640770, 13526547, 27564804, 56183565, 114612879, 234187293, 479442918, 983236998, 2018936664, 4149222198
Offset: 0
-
a(n) = sum(k=0, n\5, binomial(n-4*k, k)*binomial(n-2*k+1, n-4*k)/(n-2*k+1));
A379462
a(n) is the total number of paths starting at (0, 0), ending at (n, 0), consisting of steps (1, 1), (1, 0), (1, -2), and staying on or above y = -3.
Original entry on oeis.org
1, 1, 1, 4, 13, 31, 75, 204, 561, 1499, 4001, 10814, 29364, 79704, 216672, 590764, 1614421, 4419049, 12116139, 33277722, 91546143, 252209535, 695803659, 1922166420, 5316714156, 14723570406, 40820144106, 113293243636, 314759548879, 875342190283, 2436582442381
Offset: 0
For n = 3, the a(3)=4 paths are DUU, HHH, UDU, UUD, where U=(1,1), D=(1,-2) and H=(1,0). An example of a path with these steps, but not staying on or above y = -3, is for n=6: DDUUUU.
-
lista(nn) = my(v=vector(nn+5), w); print1(v[4]=1); for(n=1, nn, w=v; for(i=1, n+3, w[i]+=v[i+2]; w[i+1]+=v[i]); v=w; print1(", ", v[4])); \\ Jinyuan Wang, Jan 07 2025