login
This site is supported by donations to The OEIS Foundation.

 

Logo

The OEIS is looking to hire part-time people to help edit core sequences, upload scanned documents, process citations, fix broken links, etc. - Neil Sloane, njasloane@gmail.com

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A047998 Triangle of numbers a(n,k) = number of "fountains" with n coins, k in the bottom row. 6
1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 2, 1, 0, 0, 0, 1, 3, 1, 0, 0, 0, 1, 3, 4, 1, 0, 0, 0, 0, 3, 6, 5, 1, 0, 0, 0, 0, 2, 7, 10, 6, 1, 0, 0, 0, 0, 1, 7, 14, 15, 7, 1, 0, 0, 0, 0, 1, 5, 17, 25, 21, 8, 1, 0, 0, 0, 0, 0, 5, 16, 35, 41, 28, 9, 1, 0, 0, 0, 0, 0, 3, 16, 40, 65, 63, 36, 10, 1, 0, 0, 0, 0, 0, 2, 14, 43, 86, 112, 92, 45, 11, 1, 0, 0, 0, 0, 0, 1, 11, 44, 102, 167, 182, 129, 55, 12, 1, 0, 0, 0, 0, 0, 1, 9, 40, 115, 219, 301, 282, 175, 66, 13, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,14

REFERENCES

R. K. Guy, personal communication to N. J. A. Sloane.

See A005169 for further references.

LINKS

Alois P. Heinz, Rows n = 0..132, flattened

A. M. Odlyzko and H. S. Wilf, The editor's corner: n coins in a fountain, Amer. Math. Monthly, 95 (1988), 840-843.

FORMULA

G.f.: 1/(1 - y*x / (1 - y*x^2 / (1 - y*x^3 / ( ... )))), from the Odlyzko/Wilf reference. [Joerg Arndt, Mar 25 2014]

EXAMPLE

Triangle begins:

00:  1;

01:  0,1;

02:  0,0,1;

03:  0,0,1,1;

04:  0,0,0,2,1;

05:  0,0,0,1,3,1;

06:  0,0,0,1,3,4,1;

07:  0,0,0,0,3,6,5,1;

08:  0,0,0,0,2,7,10,6,1;

09:  0,0,0,0,1,7,14,15,7,1;

10:  0,0,0,0,1,5,17,25,21,8,1;

11:  0,0,0,0,0,5,16,35,41,28,9,1;

12:  0,0,0,0,0,3,16,40,65,63,36,10,1;

13:  0,0,0,0,0,2,14,43,86,112,92,45,11,1;

14:  0,0,0,0,0,1,11,44,102,167,182,129,55,12,1;

15:  0,0,0,0,0,1,9,40,115,219,301,282,175,66,13,1;

16:  0,0,0,0,0,0,7,37,118,268,434,512,420,231,78,14,1;

17:  0,0,0,0,0,0,5,32,118,303,574,806,831,605,298,91,15,1;

...

From Joerg Arndt, Mar 25 2014: (Start)

The compositions (compositions starting with part 1 and up-steps <= 1) corresponding to row n=8 with their base lengths are:

01:    [ 1 2 3 2 ]               4

02:    [ 1 2 2 3 ]               4

03:    [ 1 2 3 1 1 ]             5

04:    [ 1 2 2 2 1 ]             5

05:    [ 1 1 2 3 1 ]             5

06:    [ 1 2 2 1 2 ]             5

07:    [ 1 2 1 2 2 ]             5

08:    [ 1 1 2 2 2 ]             5

09:    [ 1 1 1 2 3 ]             5

10:    [ 1 2 2 1 1 1 ]           6

11:    [ 1 2 1 2 1 1 ]           6

12:    [ 1 1 2 2 1 1 ]           6

13:    [ 1 2 1 1 2 1 ]           6

14:    [ 1 1 2 1 2 1 ]           6

15:    [ 1 1 1 2 2 1 ]           6

16:    [ 1 2 1 1 1 2 ]           6

17:    [ 1 1 2 1 1 2 ]           6

18:    [ 1 1 1 2 1 2 ]           6

19:    [ 1 1 1 1 2 2 ]           6

20:    [ 1 2 1 1 1 1 1 ]         7

21:    [ 1 1 2 1 1 1 1 ]         7

22:    [ 1 1 1 2 1 1 1 ]         7

23:    [ 1 1 1 1 2 1 1 ]         7

24:    [ 1 1 1 1 1 2 1 ]         7

25:    [ 1 1 1 1 1 1 2 ]         7

26:    [ 1 1 1 1 1 1 1 1 ]       8

There are none with base length <= 3, two with base length 4, etc., giving row 8 [0,0,0,0,2,7,10,6,1]

(End)

MATHEMATICA

max = 17; f = 1/Fold[1 - y*x^#2/#1&, 1, Range[max] // Reverse]; s = Series[f, {x, 0, max+1}, {y, 0, max+1}]; a[n_, k_] := SeriesCoefficient[s, {x, 0, n}, {y, 0, k}]; a[0, 0] = 1; Table[a[n, k], {n, 0, max}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 24 2015 *)

PROG

(PARI)

N=22; x='x+O('x^N);

G(k)=if (k>N, 1, 1/(1-y*x^k*G(k+1)));

V=Vec( G(1) );

my( N=#V );

rvec(V) = { V=Vec(V); my(n=#V); vector(n, j, V[n+1-j] ); }

for(n=1, N, print( rvec( V[n]) ) ); \\ print triangle

\\ Joerg Arndt, Mar 25 2014

CROSSREFS

Row sums give A005169 (set x=1 in the g.f.).

Column sums give A000108 (set y=1 in the g.f.). [Joerg Arndt, Mar 25 2014]

T(2n+1,n+1) gives A058300(n). - Alois P. Heinz, Jun 24 2015

Sequence in context: A057094 A284938 A186084 * A017847 A127841 A091006

Adjacent sequences:  A047995 A047996 A047997 * A047999 A048000 A048001

KEYWORD

nonn,tabl,easy,nice

AUTHOR

N. J. A. Sloane.

EXTENSIONS

More terms by Joerg Arndt, Mar 08 2011

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified May 27 22:57 EDT 2017. Contains 287210 sequences.