A136438 Hypertribonacci number array read by antidiagonals.

0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 2, 2, 0, 0, 1, 3, 4, 4, 0, 0, 1, 4, 7, 8, 7, 0, 0, 1, 5, 11, 15, 15, 13, 0, 0, 1, 6, 16, 26, 30, 28, 24, 0, 0, 1, 7, 22, 42, 56, 58, 52, 44, 0, 0, 1, 8, 29, 64, 98, 114, 110, 96, 81, 0, 0, 1, 9, 37, 93, 162, 212, 224, 206, 177, 149

Offset

1,14

Comments

The hypertribonacci numbers are to the hyperfibonacci array of A136431 as the tribonacci numbers A000073 are to the Fibonacci numbers A000045.

Links

Table of n, a(n) for n=1..78.

Formula

a(k,n) = apply partial sum operator k times to tribonacci numbers A000073.

M. F. Hasler notes that the 8th column = vector(25,n,binomial(n+5,6)+binomial(n+5,4)+2*binomial(n+3,1)). R. J. Mathar points out that the repeated partial sums are quickly computed from their o.g.f.s (-1)^(k+1)*x^2/(-1+x+x^2+x^3)/(-1+x)^k, k=1,2,3,...

Example

The array a(k,n) begins:
========================================
n=0..|.1.|.2.|...3.|..4.|...5.|....6.|...7..|.....8.|.....9.|....10.|
========================================
k=0..|.0.|.0.|...1.|..2.|...4.|....7.|..13..|....24.|....44.|....81.| A000073
k=1..|.0.|.0.|...2.|..4.|...8.|...15.|..28..|....52.|....96.|...177.| A008937
k=2..|.0.|.0.|...3.|..7.|..15.|...30.|..58..|...110.|...206.|...383.| A062544
k=3..|.0.|.0.|...4.|.11.|..26.|...56.|..114.|...224.|...430.|...813.|
k=4..|.0.|.0.|...5.|.16.|..42.|...98.|..212.|...436.|...866.|..1679.|
k=5..|.0.|.0.|...6.|.22.|..64.|..162.|..374.|...810.|..1676.|..3355.|
k=6..|.0.|.0.|...7.|.29.|..93.|..255.|..629.|..1439.|..3115.|..6470.|
k=7..|.0.|.0.|...8.|.37.|.130.|..385.|.1014.|..2453.|..5568.|.12038.|
k=8..|.0.|.0.|...9.|.46.|.176.|..561.|.1575.|..4028.|..9596.|.21634.|
k=9..|.0.|.0.|..10.|.56.|.232.|..793.|.2368.|..6396.|.15992.|.37626.|
k=10.|.0.|.0.|..11.|.67.|.299.|.1092.|.3460.|..9856.|.25848.|.63474.|
========================================

Prog

(PARI) \ create the n X n matrix of nonzero values
hypertribo(n)={ local(M=matrix(n, n)); M[1, ]=Vec(1/(1-x-x^2-x^3)+O(x^n));
M[, 1]=vector(n, i, 1)~; for(i=2, n, for(j=2, n, M[i, j]=M[i-1, j]+M[i, j-1])); M}
Example: gp> hypertribo(10)
[1 1 2 4 7 13 24 44 81 149]
[1 2 4 8 15 28 52 96 177 326]
[1 3 7 15 30 58 110 206 383 709]
[1 4 11 26 56 114 224 430 813 1522]
[1 5 16 42 98 212 436 866 1679 3201]
[1 6 22 64 162 374 810 1676 3355 6556]
[1 7 29 93 255 629 1439 3115 6470 13026]
[1 8 37 130 385 1014 2453 5568 12038 25064]
[1 9 46 176 561 1575 4028 9596 21634 46698]
[1 10 56 232 793 2368 6396 15992 37626 84324]
/* create the sequence: "...read by antidiagonals" )*/
hypertriboantidiag(n)={n=hypertribo(n); concat(vector(#n, i, vector(i, j, n[j, i-j+1])))}
Example: gp> hypertriboantidiag(10) /* Comment: any 1 except a(2) marks the end of antidiagonal */
[1, 1, 1, 2, 2, 1, 4, 4, 3, 1, 7, 8, 7, 4, 1, 13, 15, 15, 11, 5, 1,
24, 28, 30, 26, 16, 6, 1, 44, 52, 58, 56, 42, 22, 7, 1, 81, 96, 110,
114, 98, 64, 29, 8, 1, 149, 177, 206, 224, 212, 162, 93, 37, 9, 1]

Crossrefs

n=4 column = A000124

n=5 column = A000125

n=6 column = A055795

Cf. A000045, A000073, A000124, A000125, A008937, A055795, A062544, A136431, A137176.

Sequence in context: A035379 A280452 A096587 * A370063 A059848 A352361

Adjacent sequences: A136435 A136436 A136437 * A136439 A136440 A136441

Keyword

easy,nonn,tabl

Author

Jonathan Vos Post, Apr 13 2008

Extensions

Examples corrected by R. J. Mathar, Apr 21 2008

Status

approved