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A136438 Hypertribonacci number array read by antidiagonals.

%I A136438 #17 Feb 15 2025 02:07:25
%S A136438 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,
%T A136438 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,
%U A136438 64,98,114,110,96,81,0,0,1,9,37,93,162,212,224,206,177,149
%N A136438 Hypertribonacci number array read by antidiagonals.
%C A136438 The hypertribonacci numbers are to the hyperfibonacci array of A136431 as the tribonacci numbers A000073 are to the Fibonacci numbers A000045.
%F A136438 a(k,n) = apply partial sum operator k times to tribonacci numbers A000073.
%F A136438 _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,...
%e A136438 The array a(k,n) begins:
%e A136438 ========================================
%e A136438 n=0..|.1.|.2.|...3.|..4.|...5.|....6.|...7..|.....8.|.....9.|....10.|
%e A136438 ========================================
%e A136438 k=0..|.0.|.0.|...1.|..2.|...4.|....7.|..13..|....24.|....44.|....81.| A000073
%e A136438 k=1..|.0.|.0.|...2.|..4.|...8.|...15.|..28..|....52.|....96.|...177.| A008937
%e A136438 k=2..|.0.|.0.|...3.|..7.|..15.|...30.|..58..|...110.|...206.|...383.| A062544
%e A136438 k=3..|.0.|.0.|...4.|.11.|..26.|...56.|..114.|...224.|...430.|...813.|
%e A136438 k=4..|.0.|.0.|...5.|.16.|..42.|...98.|..212.|...436.|...866.|..1679.|
%e A136438 k=5..|.0.|.0.|...6.|.22.|..64.|..162.|..374.|...810.|..1676.|..3355.|
%e A136438 k=6..|.0.|.0.|...7.|.29.|..93.|..255.|..629.|..1439.|..3115.|..6470.|
%e A136438 k=7..|.0.|.0.|...8.|.37.|.130.|..385.|.1014.|..2453.|..5568.|.12038.|
%e A136438 k=8..|.0.|.0.|...9.|.46.|.176.|..561.|.1575.|..4028.|..9596.|.21634.|
%e A136438 k=9..|.0.|.0.|..10.|.56.|.232.|..793.|.2368.|..6396.|.15992.|.37626.|
%e A136438 k=10.|.0.|.0.|..11.|.67.|.299.|.1092.|.3460.|..9856.|.25848.|.63474.|
%e A136438 ========================================
%o A136438 (PARI) \\ create the n X n matrix of nonzero values
%o A136438 hypertribo(n)={ local(M=matrix(n,n)); M[1,]=Vec(1/(1-x-x^2-x^3)+O(x^n));
%o A136438 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}
%o A136438 { hypertribo(10) }
%Y A136438 Columns n=4..6 are A000124, A000125, A055795.
%Y A136438 Cf. A000045, A000073, A008937, A062544, A136431, A137176.
%K A136438 easy,nonn,tabl
%O A136438 1,14
%A A136438 _Jonathan Vos Post_, Apr 13 2008
%E A136438 Examples corrected by _R. J. Mathar_, Apr 21 2008