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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 October 13 18:57 EDT 2019. Contains 327981 sequences. (Running on oeis4.)