A004798 -id:A004798 - cp's OEIS frontend

A185676 Riordan array (((1+x)/(1-x-x^2))^m, x*A000108(x)), m=2.

1, 4, 1, 10, 5, 1, 22, 16, 6, 1, 45, 45, 23, 7, 1, 88, 121, 76, 31, 8, 1, 167, 325, 237, 116, 40, 9, 1, 310, 895, 728, 403, 166, 50, 10, 1, 566, 2563, 2253, 1358, 630, 227, 61, 11, 1, 1020, 7670, 7104, 4541, 2288, 930, 300, 73, 12, 1, 1819, 23939, 22919, 15249, 8145, 3604, 1316, 386, 86, 13, 1

Offset: 0

Vladimir Kruchinin, Feb 09 2011

			1;
4,1;
10,5,1;
22,16,6,1;
45,45,23,7,1;
88,121,76,31,8,1;
167,325,237,116,40,9,1;
310,895,728,403,166,50,10,1;

Column k=0 gives: A004798(n+1).

r[n_, k_, m_] := k*Sum[ Sum[ Binomial[j-1, m-1]*Binomial[j, i+m-j], {j, m, i+m}]*Binomial[2*(n-i)-k-1, n-i-1]/(n-i), {i, 0, n-k}]; r[n_, 0, m_] := Sum[ Binomial[i-1, m-1]*Binomial[i, n+m-i], {i, m, n+m}]; Table[r[n, k, 2], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 21 2013 *)

R(n,k,m) = k*sum(i=0..n-k, sum(j=m..i+m, binomial(j-1,m-1) * binomial(j,i+m-j)) * binomial(2*(n-i)-k-1,n-i-1)/(n-i)), k>0, m=2; R(n,0,m) = sum(i=m..n+m, binomial(i-1,m-1) * binomial(i,n+m-i)).

A210637 Triangle T(n,k), read by rows, given by (2, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 2, 2, 5, 8, 3, 12, 27, 20, 5, 29, 84, 91, 44, 8, 70, 248, 352, 251, 90, 13, 169, 708, 1240, 1164, 618, 176, 21, 408, 1973, 4106, 4771, 3344, 1414, 334, 34, 985, 5400, 13010, 18000, 15645, 8748, 3073, 620, 55

Offset: 0

Philippe Deléham, Mar 26 2012

Row sums are powers of 4 (A000302).

			Triangle begins :
1
2, 2
5, 8, 3
12, 27, 20, 5
29, 84, 91, 44, 8
70, 248, 352, 251, 90, 13
169, 708, 1240, 1164, 618, 176, 21
408, 1973, 4106, 4771, 3344, 1414, 334, 34
985, 5400, 13010, 18000, 15645, 8748, 3073, 620, 55
2378, 14574, 39880, 63966, 66282, 46014, 21400, 6429, 1132, 89
5741, 38896, 119129, 217232, 261185, 216348, 125028, 49772, 13061, 2040, 144

Cf. A000045, A000129, A000302, A261056 (2nd column).

G.f.: (1+y*x)/(1-(y+2)*x-(y+1)^2*x^2).
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) + T(n-2,k) + 2*T(n-2,k-1) + T(n-2,k-2), T(0,0) = 1, T(1,0) = T(1,1) = 2 and T(n,k) = 0 if k<0 or if k>n.
Sum_{k, 0<=k<=n} T(n,k)*x^k = (-1)^n*A159612(n+1), (-1)^n*A000034(n), A000007(n), A000129(n+1), A000302(n) for x = -3, -2, -1, 0, 1 respectively.
T(n,0) = A000129(n+1), T(n,n) = A000045(n+2), T(n+1,n) = 2*A004798(n+1).

A379037 G.f. A(x) satisfies A(x) = ( (1 + x) * (1 + x*A(x)^(3/2)) )^2.

Original entry on oeis.org

1, 4, 18, 106, 689, 4782, 34707, 260190, 1999168, 15660176, 124596498, 1004110948, 8179379807, 67239070868, 557098881919, 4647368670950, 39001655222787, 329048378867468, 2789241880512898, 23743798316713368, 202894843070927859, 1739775692700850554

Offset: 0

Seiichi Manyama, Dec 14 2024

nonn

Cf. A004798, A073155, A379039.

Cf. A364336, A366200.

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

G.f.: A(x) = B(x)^2 where B(x) is the g.f. of A364336.

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

A130138 Triangle read by rows: T(n,k) is the number of Fibonacci binary words of length n and having k 1011's (n>=0, 0<=k<=floor((n-1)/3)). A Fibonacci binary word is a binary word having no 00 subword.

Original entry on oeis.org

1, 2, 3, 5, 7, 1, 9, 4, 11, 10, 13, 20, 1, 15, 35, 5, 17, 56, 16, 19, 84, 40, 1, 21, 120, 86, 6, 23, 165, 166, 23, 25, 220, 296, 68, 1, 27, 286, 496, 171, 7, 29, 364, 791, 382, 31, 31, 455, 1211, 781, 105, 1, 33, 560, 1792, 1488, 300, 8, 35, 680, 2576, 2678, 756, 40, 37

Offset: 0

Emeric Deutsch, May 13 2007

Row n has 1+floor((n-1)/3) terms. Row sums are the Fibonacci numbers (A000045). T(n,0)=A004280(n+1). Sum(k*T(n,k), k>=0)=A004798(n-3) (n>=4).

			T(7,2)=1 because we have 1011011.
Triangle starts:
1;
2;
3;
5;
7,1;
9,4;
11,10;
13,20,1;
15,35,5;

Cf. A000045, A004280, A004798.

G:=(1+z)*(1+z^3-t*z^3)/(1-z-z^2+z^3-t*z^3): Gser:=simplify(series(G,z=0,24)): for n from 0 to 21 do P[n]:=sort(coeff(Gser,z,n)) od: 1; for n from 1 to 21 do seq(coeff(P[n],t,j),j=0..floor((n-1)/3)) od; # yields sequence in triangular form

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

cp's OEIS Frontend

A185676 Riordan array (((1+x)/(1-x-x^2))^m, x*A000108(x)), m=2.

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A210637 Triangle T(n,k), read by rows, given by (2, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

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A379037 G.f. A(x) satisfies A(x) = ( (1 + x) * (1 + x*A(x)^(3/2)) )^2.

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A130138 Triangle read by rows: T(n,k) is the number of Fibonacci binary words of length n and having k 1011's (n>=0, 0<=k<=floor((n-1)/3)). A Fibonacci binary word is a binary word having no 00 subword.

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