The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation. Thanks to everyone who made a donation during our annual appeal!
To see the list of donors, or make a donation, see the OEIS Foundation home page.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A115728 Number of subpartitions of partitions in Abramowitz and Stegun order. 26
 1, 2, 3, 3, 4, 5, 4, 5, 7, 6, 7, 5, 6, 9, 9, 10, 9, 9, 6, 7, 11, 12, 13, 10, 14, 13, 10, 12, 11, 7, 8, 13, 15, 16, 14, 19, 17, 16, 16, 19, 16, 14, 15, 13, 8, 9, 15, 18, 19, 18, 24, 21, 15, 23, 22, 26, 21, 19, 22, 23, 24, 19, 15, 18, 18, 15, 9, 10, 17, 21, 22, 22, 29 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS subpart([n^k]) = C(n+k,k); subpart([1,2,3,...,n]) = C_n = A000108(n). The b(i,j) defined in the formula for sequences [1,2,3,...] form A009766. LINKS M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. FORMULA For a partition P = [p_1,...,p_n] with the p_i in increasing order, define b(i,j) to be the number of subpartitions of [p_1,...,p_i] with the i-th part = j (b(i,0) is subpartitions with less than i parts). Then b(1,j)=1 for j<=p_1, b(i+1,j) = Sum_{k=0}^j} b(i,k) for 0<=j<=p_{i+1}; and the total number of subpartitions is sum_{k=1}^{p_n} b(n,k). For a partition P = {p(n)}, the number of subpartitions {s(n)} of P can be determined by the g.f.: 1/(1-x) = Sum_{n>=0} s(n)*x^n*(1-x)^p(n). - Paul D. Hanna, Jul 03 2006 EXAMPLE Partition 5 in A&S order is [2,1]; it has 5 subpartitions: [], , , [1^2] and [2,1] itself. PROG (PARI) /* Expects input as vector in increasing order - e.g. [1, 1, 2, 3] */ subpart(p)=local(i, j, v, n); n=matsize(p); if(n==0, 1, v=vector(p[n]+1); v =1; for(i=1, n, for(j=1, p[i], v[j+1]+=v[j])); for(j=1, p[n], v[j+1]+=v[j]); v[p[n ]+1]) (PARI) /* Given Partition p(), Find Subpartitions s(): */ {s(n)=polcoeff(x^n-sum(k=0, n-1, s(k)*x^k*(1-x+x*O(x^n))^p(k)), n)} - Paul D. Hanna, Jul 03 2006 CROSSREFS Cf. A115729, A036036, A000108, A009766, A007318. Sequence in context: A305579 A321440 A115729 * A188553 A026354 A179840 Adjacent sequences:  A115725 A115726 A115727 * A115729 A115730 A115731 KEYWORD nonn AUTHOR Franklin T. Adams-Watters, Mar 11 2006 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
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified January 19 00:40 EST 2020. Contains 331030 sequences. (Running on oeis4.)