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 A107354 To compute a(n) we first write down 2^n 1's in a row. Each row takes the right half of the previous row and each element in it equals sum of the elements in the previous row starting at the middle. The single element in the last row is a(n). 14
 1, 1, 2, 7, 44, 516, 11622, 512022, 44588536, 7718806044, 2664170119608, 1836214076324153, 2529135272371085496, 6964321029630556852944, 38346813253279804426846032, 422247020982575523983378003936, 9298487213328788062025571134762096 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Number of subpartitions of partition [1,3,7,...,2^n-1]. - Franklin T. Adams-Watters, Mar 11 2006 Can also be computed summing forwards: 1 1,1 1,2,2, 2 1,3,5, 7, 7, 7, 7, 7 1,4,9,16,23,30,37,44,44,44,44,44,44,44,44,44 LINKS Alois P. Heinz, Table of n, a(n) for n = 0..82 (first 26 terms from Reinhard Zumkeller) FORMULA a(n) = C(2^(n-1)+n-2,n-1) - Sum_{k=1..n-2} a(k)*C(2^(n-1)-2^k+n-k-1,n-k) for n>=2, with a(0)=1, a(1)=1, where C = binomial. - Paul D. Hanna, May 24 2005 The first number in row 3 is 2^(n-2)+1. - Ralf Stephan, May 24 2005 G.f.: 1/(1-x) = Sum_{n>=0} a(n)*x^n*(1-x)^(2^n-1) (g.f. of subpartitions). - Paul D. Hanna, Jul 03 2006 G.f.: 1 = Sum_{n>=0} a(n)*x^n/(1+x)^(2^n+n). - Paul D. Hanna, Jul 03 2006 EXAMPLE For n=4, the array looks like this: 1..1..1..1..1..1..1..1..1..1..1..1..1..1..1..1 ........................1..2..3..4..5..6..7..8 ....................................5.11.18.26 .........................................18.44 ............................................44 Therefore a(4)=44. For n=5, we can illustrate the recurrence by: a(5) = 516 = C(19, 4) - ( 1*C(17, 4) + 2*C(14, 3) + 7*C(9, 2) ) = C(16+4-1, 4) - ( 1*C(16-2+4-1, 4) + 2*C(16-4+3-1, 3) + 7*C(16-8+2-1, 2) ). MAPLE a:= proc(n) option remember; `if`(n=0, 1, -add( a(j)*(-1)^(n-j)*binomial(2^j, n-j), j=0..n-1)) end: seq(a(n), n=0..16); # Alois P. Heinz, Jul 08 2022 MATHEMATICA f[n_] := If[n == 0, 1, Binomial[2^(n - 1) + n - 2, n - 1] - Sum[ f[k]*Binomial[2^(n - 1) - 2^k + n - k - 1, n - k], {k, n - 2}]]; Table[ f[n], {n, 0, 15}] (* Robert G. Wilson v, May 25 2005 *) Table[NestWhile[Accumulate[Drop[#, Ceiling[Length[#]/2]]]&, PadRight[{}, 2^n+1, 1], Length[ #]> 1&], {n, 0, 16}]//Flatten (* Harvey P. Dale, Jun 24 2018 *) PROG (PARI) {a(n)=if(n==0, 1, binomial(2^(n-1)+n-2, n-1)- sum(k=1, n-2, a(k)*binomial(2^(n-1)-2^k+n-k-1, n-k)))} \\ Paul D. Hanna, May 24 2005 (PARI) {a(n)=polcoeff(x^n-sum(k=0, n-1, a(k)*x^k*(1-x+x*O(x^n))^(2^k-1) ), n)} \\ Paul D. Hanna, May 24 2005 (Haskell) a107354 n = head \$ snd \$ until ((== 1) . fst) f (2^n, replicate (2^n) 1) where f (len, xs) = (len', scanl1 (+) \$ drop len' xs) where len' = len `div` 2 -- Feasible only for small n. -- Reinhard Zumkeller, Nov 20 2011 CROSSREFS Cf. A105996; variants: A109055 - A109061; subpartitions defined: A115728, A115729. Column k=2 of A355576. Sequence in context: A355109 A278295 A356613 * A006118 A083670 A270357 Adjacent sequences: A107351 A107352 A107353 * A107355 A107356 A107357 KEYWORD nonn,nice AUTHOR Max Alekseyev, May 24 2005 EXTENSIONS Edited by Paul D. Hanna, Jul 03 2006 STATUS approved

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Last modified June 21 07:08 EDT 2024. Contains 373540 sequences. (Running on oeis4.)