1,14

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

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

Maximilian Hasler notes that the 8-th 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,...

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.|

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(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]

n=4 column = A000124

n=5 column = A000125

n=6 column = A055795

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

Adjacent sequences: A136435 A136436 A136437 this_sequence A136439 A136440 A136441

Sequence in context: A029392 A035379 A096587 this_sequence A059848 A036865 A125226

easy,nonn,tabl

Jonathan Vos Post (jvospost3(AT)gmail.com), Apr 13 2008

Examples corrected by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Apr 21 2008