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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
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. (1-((-1)^T(n, k)))/2 = A051731, see formula by Hieronymus Fischer in A022003.
Sequence in context: A055794 A092905 A052509 * A228125 A227588 A093628
KEYWORD
nonn,tabl
AUTHOR
Mats Granvik, Jan 26 2010
STATUS
approved

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