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 A125714 Alfred Moessner's factorial triangle. 12

%I

%S 1,2,3,6,11,6,24,50,35,10,120,274,225,85,15,720,1764,1624,735,175,21,

%T 5040,13068,13132,6769,1960,322,28,40320,109584,118124,67284,22449,

%U 4536,546,36,362880,1026576,1172700,723680,269325,63273,9450,870,45,3628800

%N Alfred Moessner's factorial triangle.

%C Row sums of the triangle = 1, 5, 23, 119, 719...(matching the terms 0, 0, 1, 5, 23, 119, 719...; of A033312).

%C The name of the triangle derives from the fact that A125714(A000124(n)) = A000142(n) for n > 0. Moessner's method uses only additions to compute the factorial n!. - _Peter Luschny_, Jan 27 2009

%D J. H. Conway and R. K. Guy, "The Book of Numbers", Springer-Verlag, 1996, p. 64 (based on the work of Alfred Moessner).

%D Alfred Moessner, Eine Bemerkung ueber die Potenzen der natuerlichen Zahlen. S.-B. Math.-Nat. Kl. Bayer. Akad. Wiss., 29, 1951.

%D Oskar Perron, Beweis des Moessnerschen Satzes. S.-B. Math.-Nat. Kl. Bayer. Akad. Wiss., 31-34, 1951.

%H Joshua Zucker, <a href="/A125714/b125714.txt">Table of n, a(n) for n = 1..66</a>

%F Starting with the natural numbers, circle each triangular number. Underneath, take partial sums of the uncircled terms and circle the terms in this row which are offset one place to the left of the circled 1, 3, 6, 10...in the first row. Repeat with analogous operations in succeeding rows. The circled terms in the infinite set become the triangle.

%e An "x" prefaced before each term will indicate the term following the x being circled.

%e x1 2 x3 4 5 x6 7 8 9 x10 11 12 13 14 x15...

%e __x2 6 x11 18 26 x35 46 58 71 x85...

%e _____________x6 24 x50 96 154 x225...

%e _________________________x24 120 x274...

%e ___________________________________________x120...

%e ...

%e i.e. circle the triangular terms in row 1. In row 2, take partial sums of the uncircled terms and circle the terms offset one place to the left of the triangular terms in row 1. Continue in subsequent rows with analogous operations. The triangle consists of the infinite set of terms prefaced with the x (circled on page 64 of "The Book of Numbers".).

%p Contribution from _Peter Luschny_, Jan 27 2009: (Start)

%p a := proc(n) local s,m,k,i; s := array(0..n); s := 1;

%p for m from 1 to n do s[m] := 0; for k from m by - 1 to 1 do

%p for i from 1 to k do s[i] := s[i] + s[i - 1] od; lprint(s[k]);

%p if k = n then RETURN(s[n]) fi od; lprint("-") od end: (End)

%t n = 10; A125714 = Reap[ ClearAll[s]; s = 1; For[m = 1, m <= n, m++, s[m] = 0; For[k = m, k >= 1, k--, For[i = 1, i <= k, i++, s[i] = s[i] + s[i-1]]; Sow[s[k]]; If[k == n, Print[n, "! = ", s[n]]; Break[]]]]][[2, 1]] (* _Jean-François Alcover_, Jun 29 2012, after _Peter Luschny_ *)

%o (PARI) T(n, k)={ my( s=vector(n)); for( m=1, n, forstep( j=m,1,-1, s++; for( i=2, j, s[i] += s[i-1]));

%o k<0 && print(vecextract(s,Str(m"..1"))));

%o if( k>0,s[n+1-k],vecextract(s,"-1..1"))} /* returns T[n,k], or the whole n-th row if k is not given, prints row 1...n of the triangle if k<0 */ \\ M. F. Hasler, Dec 03 2010

%Y Cf. A033312.

%K nonn,tabl

%O 1,2

%A _Gary W. Adamson_, Dec 01 2006

%E More terms from _Joshua Zucker_, Jun 17 2007

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Last modified June 24 17:58 EDT 2019. Contains 324330 sequences. (Running on oeis4.)