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}
{ hypertribo(10) }

Crossrefs

Columns n=4..6 are A000124, A000125, A055795.

Cf. A000045, A000073, A008937, A062544, A136431, A137176.

Sequence in context: A035379 A280452 A096587 * A370063 A059848 A352361

Adjacent sequences: A136435 A136436 A136437 * A136439 A136440 A136441

Keyword

easy,nonn,tabl,changed

Author

Jonathan Vos Post, Apr 13 2008

Extensions

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

Status

approved