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 A246596 Run Length Transform of Catalan numbers A000108. 11
 1, 1, 1, 2, 1, 1, 2, 5, 1, 1, 1, 2, 2, 2, 5, 14, 1, 1, 1, 2, 1, 1, 2, 5, 2, 2, 2, 4, 5, 5, 14, 42, 1, 1, 1, 2, 1, 1, 2, 5, 1, 1, 1, 2, 2, 2, 5, 14, 2, 2, 2, 4, 2, 2, 4, 10, 5, 5, 5, 10, 14, 14, 42, 132, 1, 1, 1, 2, 1, 1, 2, 5, 1, 1, 1, 2, 2, 2, 5, 14, 1, 1, 1, 2, 1, 1, 2, 5 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS The Run Length Transform of a sequence {S(n), n>=0} is defined to be the sequence {T(n), n>=0} given by T(n) = Product_i S(i), where i runs through the lengths of runs of 1's in the binary expansion of n. E.g. 19 is 10011 in binary, which has two runs of 1's, of lengths 1 and 2. So T(19) = S(1)*S(2). T(0)=1 (the empty product). LINKS Chai Wah Wu, Table of n, a(n) for n = 0..8192 FORMULA a(n) = A069739(A005940(1+n)). - Antti Karttunen, May 29 2017 EXAMPLE From Omar E. Pol, Feb 15 2015: (Start) Written as an irregular triangle in which row lengths are the terms of A011782: 1; 1; 1,2; 1,1,2,5; 1,1,1,2,2,2,5,14; 1,1,1,2,1,1,2,5,2,2,2,4,5,5,14,42; 1,1,1,2,1,1,2,5,1,1,1,2,2,2,5,14,2,2,2,4,2,2,4,10,5,5,5,10,14,14,42,132; ... Right border gives the Catalan numbers. This is simply a restatement of the theorem that this sequence is the Run Length Transform of A000108. (End) MAPLE Cat:=n->binomial(2*n, n)/(n+1); ans:=[]; for n from 0 to 100 do lis:=[]; t1:=convert(n, base, 2); L1:=nops(t1); out1:=1; c:=0; for i from 1 to L1 do if out1 = 1 and t1[i] = 1 then out1:=0; c:=c+1; elif out1 = 0 and t1[i] = 1 then c:=c+1; elif out1 = 1 and t1[i] = 0 then c:=c; elif out1 = 0 and t1[i] = 0 then lis:=[c, op(lis)]; out1:=1; c:=0; fi; if i = L1 and c>0 then lis:=[c, op(lis)]; fi; od: a:=mul(Cat(i), i in lis); ans:=[op(ans), a]; od: ans; MATHEMATICA f = CatalanNumber; Table[Times @@ (f[Length[#]]&) /@ Select[ Split[ IntegerDigits[n, 2]], #[[1]] == 1&], {n, 0, 87}] (* Jean-François Alcover, Jul 11 2017 *) PROG (Python) from operator import mul from functools import reduce from gmpy2 import divexact from re import split def A246596(n):     s, c = bin(n)[2:], [1, 1]     for m in range(1, len(s)):         c.append(divexact(c[-1]*(4*m+2), (m+2)))     return reduce(mul, (c[len(d)] for d in split('0+', s))) if n > 0 else 1 # Chai Wah Wu, Sep 07 2014 (Sage) # uses[RLT from A246660] A246596_list = lambda len: RLT(lambda n: binomial(2*n, n)/(n+1), len) A246596_list(88) # Peter Luschny, Sep 07 2014 (Scheme) ; using MIT/GNU Scheme (define (A246596 n) (fold-left (lambda (a r) (* a (A000108 r))) 1 (bisect (reverse (binexp->runcount1list n)) (- 1 (modulo n 2))))) (define A000108 (EIGEN-CONVOLUTION 1 *)) ;; Note: EIGEN-CONVOLUTION can be found from my IntSeq-library and other functions are as in A227349. - Antti Karttunen, Sep 08 2014 CROSSREFS Cf. A000108. Cf. A003714 (gives the positions of ones). Run Length Transforms of other sequences: A005940, A069739, A071053, A227349, A246588, A246595, A246660, A246661, A246674. Sequence in context: A326566 A213953 A000361 * A135723 A125311 A127568 Adjacent sequences:  A246593 A246594 A246595 * A246597 A246598 A246599 KEYWORD nonn AUTHOR N. J. A. Sloane, Sep 06 2014 STATUS approved

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Last modified May 31 02:51 EDT 2020. Contains 334747 sequences. (Running on oeis4.)