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 A172119 Sum the k preceding elements in the same column and add 1 every time. 15
 1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 4, 4, 2, 1, 1, 5, 7, 4, 2, 1, 1, 6, 12, 8, 4, 2, 1, 1, 7, 20, 15, 8, 4, 2, 1, 1, 8, 33, 28, 16, 8, 4, 2, 1, 1, 9, 54, 52, 31, 16, 8, 4, 2, 1, 1, 10, 88, 96, 60, 32, 16, 8, 4, 2, 1, 1, 11, 143, 177, 116, 63, 32, 16, 8, 4, 2, 1, 1, 12, 232, 326, 224, 124, 64, 32, 16 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS Columns are related to Fibonacci n-step numbers. Are there closed forms for the sequences in the columns? We denote by a(n,k) the number which is in the (n+1)-th row and (k+1)-th-column. With help of the definition, we also have the recurrence relation: a(n+k+1, k) = 2*a(n+k, k) - a(n, k). We see on the main diagonal the numbers 1,2,4, 8, ..., which is clear from the formula for the general term d(n)=2^n. - Richard Choulet, Jan 31 2010 Most of the paper by Dunkel (1925) is a study of the columns of this table. - Petros Hadjicostas, Jun 14 2019 LINKS Table of n, a(n) for n=0..86. O. Dunkel, Solutions of a probability difference equation, Amer. Math. Monthly, 32 (1925), 354-370; see p. 356. T. Langley, J. Liese, and J. Remmel, Generating Functions for Wilf Equivalence Under Generalized Factor Order, J. Int. Seq. 14 (2011), # 11.4.2. Eric Weisstein's World of Mathematics, Fibonacci n-Step Number. Wikipedia, Fibonacci number. FORMULA T(n,0) = 1. T(n,1) = n. T(n,2) = A000071(n+1). T(n,3) = A008937(n-2). The general term in the n-th row and k-th column is given by: a(n, k) = Sum_{j=0..floor(n/(k+1))} ((-1)^j binomial(n-k*j,n-(k+1)*j)*2^(n-(k+1)*j)). For example: a(5,3) = binomial(5,5)*2^5 - binomial(2,1)*2^1 = 28. The generating function of the (k+1)-th column satisfies: psi(k)(z)=1/(1-2*z+z^(k+1)) (for k=0 we have the known result psi(0)(z)=1/(1-z)). - Richard Choulet, Jan 31 2010 [By saying "(k+1)-th column" the author actually means "k-th column" for k = 0, 1, 2, ... - Petros Hadjicostas, Jul 26 2019] EXAMPLE Triangle begins: n\k|....0....1....2....3....4....5....6....7....8....9...10 ---|------------------------------------------------------- 0..|....1 1..|....1....1 2..|....1....2....1 3..|....1....3....2....1 4..|....1....4....4....2....1 5..|....1....5....7....4....2....1 6..|....1....6...12....8....4....2....1 7..|....1....7...20...15....8....4....2....1 8..|....1....8...33...28...16....8....4....2....1 9..|....1....9...54...52...31...16....8....4....2....1 10.|....1...10...88...96...60...32...16....8....4....2....1 MAPLE for k from 0 to 20 do for n from 0 to 20 do b(n):=sum((-1)^j*binomial(n-k*j, n-(k+1)*j)*2^(n-(k+1)*j), j=0..floor(n/(k+1))):od: seq(b(n), n=0..20):od; # Richard Choulet, Jan 31 2010 A172119 := proc(n, k) option remember; if k = 0 then 1; elif k > n then 0; else 1+add(procname(n-k+i, k), i=0..k-1) ; end if; end proc: seq(seq(A172119(n, k), k=0..n), n=0..12) ; # R. J. Mathar, Sep 16 2017 MATHEMATICA T[_, 0] = 1; T[n_, n_] = 1; T[n_, k_] /; k>n = 0; T[n_, k_] := T[n, k] = Sum[T[n-k+i, k], {i, 0, k-1}] + 1; Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten Table[Sum[(-1)^j*2^(n-k-(k+1)*j)*Binomial[n-k-k*j, n-k-(k+1)*j], {j, 0, Floor[(n-k)/(k+1)]}], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Jul 27 2019 *) PROG (PARI) T(n, k) = if(k<0 || k>n, 0, k==1 && k==n, 1, 1 + sum(j=1, k, T(n-j, k))); for(n=1, 12, for(k=0, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Jul 27 2019 (Magma) T:= func< n, k | (&+[(-1)^j*2^(n-k-(k+1)*j)*Binomial(n-k-k*j, n-k-(k+1)*j): j in [0..Floor((n-k)/(k+1))]]) >; [[T(n, k): k in [0..n]]: n in [0..12]]; // G. C. Greubel, Jul 27 2019 (Sage) @CachedFunction def T(n, k): if (k==0 and k==n): return 1 elif (k<0 or k>n): return 0 else: return 1 + sum(T(n-j, k) for j in (1..k)) [[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Jul 27 2019 (GAP) T:= function(n, k) if k=0 and k=n then return 1; elif k<0 or k>n then return 0; else return 1 + Sum([1..k], j-> T(n-j, k)); fi; end; Flat(List([0..12], n-> List([0..n], k-> T(n, k) ))); # G. C. Greubel, Jul 27 2019 CROSSREFS Cf. A000071, A008937, A144428. Cf. (1-((-1)^T(n, k)))/2 = A051731, see formula by Hieronymus Fischer in A022003. Sequence in context: A055794 A092905 A052509 * A228125 A227588 A093628 Adjacent sequences: A172116 A172117 A172118 * A172120 A172121 A172122 KEYWORD nonn,tabl AUTHOR Mats Granvik, Jan 26 2010 STATUS approved

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Last modified May 22 06:48 EDT 2024. Contains 372743 sequences. (Running on oeis4.)