A179748 - OEIS

A179748

Triangle T(n,k) read by rows. T(n,1)=1, k > 1: T(n,k) = Sum_{i=1..k-1} T(n-i,k-1).

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 1, 2, 5, 4, 1, 1, 1, 2, 6, 9, 5, 1, 1, 1, 2, 6, 15, 14, 6, 1, 1, 1, 2, 6, 20, 29, 20, 7, 1, 1, 1, 2, 6, 23, 49, 49, 27, 8, 1, 1, 1, 2, 6, 24, 71, 98, 76, 35, 9, 1, 1, 1, 2, 6, 24, 91, 169, 174, 111, 44, 10, 1, 1, 1, 2, 6, 24, 106, 259, 343, 285, 155, 54, 11, 1

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OFFSET

1,9

COMMENTS

Recurrence is half of the recurrence for divisibility in A051731. That is, without subtracting (Sum_{i=1..k-1} T(n-i,k)).

Rows tend to factorial numbers.

Row sums are A177510.

LINKS

Table of n, a(n) for n=1..91.

FORMULA

T(n,1)=1, k > 1: T(n,k) = Sum_{i=1..k-1} T(n-i,k-1).

EXAMPLE

Triangle begins:

01: 1;

02: 1, 1;

03: 1, 1, 1;

04: 1, 1, 2, 1;

05: 1, 1, 2, 3, 1;

06: 1, 1, 2, 5, 4, 1;

07: 1, 1, 2, 6, 9, 5, 1;

08: 1, 1, 2, 6, 15, 14, 6, 1;

09: 1, 1, 2, 6, 20, 29, 20, 7, 1;

10: 1, 1, 2, 6, 23, 49, 49, 27, 8, 1;

11: 1, 1, 2, 6, 24, 71, 98, 76, 35, 9, 1;

12: 1, 1, 2, 6, 24, 91, 169, 174, 111, 44, 10, 1;

13: 1, 1, 2, 6, 24, 106, 259, 343, 285, 155, 54, 11, 1;

14: 1, 1, 2, 6, 24, 115, 360, 602, 628, 440, 209, 65, 12, 1;

15: 1, 1, 2, 6, 24, 119, 461, 961, 1230, 1068, 649, 274, 77, 13, 1;

16: 1, 1, 2, 6, 24, 120, 551, 1416, 2191, 2298, 1717, 923, 351, 90, 14, 1;

17: 1, 1, 2, 6, 24, 120, 622, 1947, 3606, 4489, 4015, 2640, 1274, 441, 104, 15, 1;

...

PROG

(Excel cell formula European dot comma style) =if(column()=1; 1; if(row()>=column(); sum(indirect(address(row()-column()+1; column()-1; 4)&":"&address(row()-1; column()-1; 4); 4)); 0))

(Sage)

@CachedFunction

def T(n, k): # A179748

if n == 0: return int(k==0);

if k == 1: return int(n>=1);

return sum( T(n-i, k-1) for i in [1..k-1] );

for n in [1..15]: print([ T(n, k) for k in [1..n] ])

# Joerg Arndt, Mar 24 2014

CROSSREFS

Cf. A175105, A051731, A179749, A179750, A000142.

Sequence in context: A054124 A144406 A238888 * A096670 A130461 A225631

Adjacent sequences: A179745 A179746 A179747 * A179749 A179750 A179751

KEYWORD

nonn,tabl

AUTHOR

Mats Granvik, Jul 26 2010

STATUS

approved