A117267
Difference row triangle of A117189.
Original entry on oeis.org
1, 1, 2, 2, 3, 5, 4, 6, 9, 14, 7, 11, 17, 26, 40, 13, 20, 31, 48, 74, 114, 24, 37, 57, 88, 136, 210, 324, 44, 68, 105, 162, 250, 386, 596, 920, 81, 125, 193, 298, 460, 710, 1096, 1692, 2612, 149, 230, 355, 548, 846, 1306, 2016, 3112, 4804, 7416
Offset: 1
Taking difference rows of A117189, we get:
1, 2, 5, 14, 40, 114, ...
1, 3, 9, 26, 74, ...
2, 6, 17, 48, ...
4, 11, 31, ...
7, 20, ...
13, ...
Reorient into the triangle:
1;
1, 2;
2, 3, 5;
4, 6, 9, 14;
7, 11, 17, 26, 40;
...
-
lista(nn) = my(va = Vec(-(x-1)^2/(-1+4*x-4*x^2+2*x^3) + O(x^(nn))), v = vector(nn)); v[1] = va; for (n=2, nn, v[n] = vector(nn-n+1, k, v[n-1][k+1] - v[n-1][k]);); my(list = List()); for (n=1, nn, my(i = n, j = 1); for (k=1, n, listput(list, v[i][j]); i--; j++;);); Vec(list); \\ Michel Marcus, Aug 10 2023
A115390
Binomial transform of tribonacci sequence A000073.
Original entry on oeis.org
0, 0, 1, 4, 12, 34, 96, 272, 772, 2192, 6224, 17672, 50176, 142464, 404496, 1148480, 3260864, 9258528, 26287616, 74638080, 211918912, 601698560, 1708394752, 4850622592, 13772308480, 39103533056, 111026143488, 315235058688, 895042726912, 2541282959872
Offset: 0
1*0 = 0.
1*0 + 1*0 = 0.
1*0 + 2*0 + 1*1 = 1.
1*0 + 3*0 + 3*1 + 1* 1 = 4.
1*0 + 4*0 + 6*1 + 4*1 + 1*2 = 12.
- Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
- J. Pan, Multiple Binomial Transforms and Families of Integer Sequences, J. Int. Seq. 13 (2010), 10.4.2
- J. Pan, Some Properties of the Multiple Binomial Transform and the Hankel Transform of Shifted Sequences, J. Int. Seq. 14 (2011) # 11.3.4, remark 14.
- Eric Weisstein's World of Mathematics, Binomial Transform.
- Index entries for linear recurrences with constant coefficients, signature (4,-4,2).
-
a115390 n = a115390_list !! n
a115390_list = 0 : 0 : 1 : map (* 2) (zipWith (-) a115390_list
(tail $ map (* 2) $ zipWith (-) a115390_list (tail a115390_list)))
-- Reinhard Zumkeller, Oct 21 2011
-
b[0]=b[1]=0;b[2]=1;b[n_]:=b[n]=b[n-1]+b[n-2]+b[n-3]; a[n_]:=Sum[n!/(k!*(n-k)!)*b[k],{k,0,n}];Table[a[n],{n,0,27}] (* Farideh Firoozbakht, Mar 11 2006 *)
-
sum(sum(binomial(j-1,k-1)*2^(j-k)*binomial(n-j+k-1,2*k-1),j,k,n-k),k,1,n); /* Vladimir Kruchinin, Aug 18 2010 */
A117268
Triangle, binomial transform of the tribonacci sequence.
Original entry on oeis.org
1, 1, 1, 2, 1, 2, 4, 2, 3, 5, 7, 4, 6, 9, 14, 13, 7, 11, 17, 26, 40, 24, 13, 20, 31, 48, 74, 114, 44, 24, 37, 57, 88, 136, 210, 324, 81, 44, 68, 105, 162, 250, 386, 596, 920
Offset: 1
Taking difference rows of A117267: (1; 1, 2; 2, 3, 5; 4, 6, 9, 14;...), we get A117268:
1;
1, 1;
2, 1, 2;
4, 2, 3, 5;
7, 4, 6, 9, 14;
13, 7, 11, 17, 26, 40;
24, 13, 20, 31, 48, 74, 114;
...
A159035
a(0)=1=a(1), a(2)=2, a(3)=5; thereafter a(n+3)=4*a(n+2)-4*a(n+1)+2*a(n) for n>=1.
Original entry on oeis.org
1, 1, 2, 5, 14, 40, 114, 324, 920, 2612, 7416, 21056, 59784, 169744, 481952, 1368400, 3885280, 11031424, 31321376, 88930368, 252498816, 716916544, 2035531648, 5779458048, 16409538688, 46591385856, 132286304768, 375598753024
Offset: 0
Showing 1-4 of 4 results.
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