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%I #24 Feb 11 2024 17:28:50
%S 1,0,1,1,2,2,2,3,4,4,4,5,6,6,7,5,7,8,9,9,9,8,11,11,12,12,13,13,13,14,
%T 15,15,16,16,17,16,17,18,19,19,20,20,21,21,22,21,23,23,24,24,25,25,26,
%U 26,27,26,26,28,29,29,30,30,31,31,32,32,32,33,34,34,34,35,36,36,37,37,38
%N Number of distinct new terms in row n of Pascal's triangle.
%C Partial sums are in A126256.
%C n occurs a(n) times in A265912. - _Reinhard Zumkeller_, Dec 18 2015
%H Nick Hobson, <a href="/A126257/b126257.txt">Table of n, a(n) for n = 0..1000</a>
%H Nick Hobson, <a href="/A126257/a126257.py.txt">Python program for this sequence</a>
%e Row 6 of Pascal's triangle is: 1, 6, 15, 20, 15, 6, 1. Of these terms, only 15 and 20 do not appear in rows 0-5. Hence a(6)=2.
%o (PARI) lim=77; z=listcreate(1+lim^2\4); print1(1, ", "); r=1; for(a=1, lim, for(b=1, a\2, s=Str(binomial(a, b)); f=setsearch(z, s, 1); if(f, listinsert(z, s, f))); print1(1+#z-r, ", "); r=1+#z)
%o (Haskell)
%o import Data.List.Ordered (minus, union)
%o a126257 n = a126257_list !! n
%o a126257_list = f [] a034868_tabf where
%o f zs (xs:xss) = (length ys) : f (ys `union` zs) xss
%o where ys = xs `minus` zs
%o -- _Reinhard Zumkeller_, Dec 18 2015
%o (Python)
%o def A126257(n):
%o if n:
%o s, c = (1,), {1}
%o for i in range(n-1):
%o c.update(set(s:=(1,)+tuple(s[j]+s[j+1] for j in range(len(s)-1))+(1,)))
%o return len(set((1,)+tuple(s[j]+s[j+1] for j in range(len(s)-1))+(1,))-c)
%o return 1 # _Chai Wah Wu_, Oct 17 2023
%Y Cf. A007318, A062854, A126254, A126255, A126256.
%Y Cf. A034868, A265912.
%K easy,nonn
%O 0,5
%A _Nick Hobson_, Dec 24 2006
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