cp's OEIS Frontend

A125714 Alfred Moessner's factorial triangle.

%I A125714 #46 Jul 27 2021 06:30:38
%S A125714 1,2,3,6,11,6,24,50,35,10,120,274,225,85,15,720,1764,1624,735,175,21,
%T A125714 5040,13068,13132,6769,1960,322,28,40320,109584,118124,67284,22449,
%U A125714 4536,546,36,362880,1026576,1172700,723680,269325,63273,9450,870,45,3628800
%N A125714 Alfred Moessner's factorial triangle.
%C A125714 Successive numbers arising from the Moessner construction of the factorial numbers. - _N. J. A. Sloane_, Jul 27 2021
%C A125714 Row sums of the triangle = 1, 5, 23, 119, 719, ...(matching the terms 0, 0, 1, 5, 23, 119, 719, ...; of A033312).
%C A125714 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
%C A125714 If n = (m^2+m+2)/2 then a(n) = (m+1)!. For example, taking m = 3, n = 7, and indeed a(7) = 4! = 24. - _N. J. A. Sloane_, Jul 27 2021
%D A125714 J. H. Conway and R. K. Guy, "The Book of Numbers", Springer-Verlag, 1996. Sequence can be seen by reading the successive circled numbers in the "factorial" section on page 64 (based on the work of Alfred Moessner).
%H A125714 Joshua Zucker, <a href="/A125714/b125714.txt">Table of n, a(n) for n = 1..66</a>
%H A125714 G. S. Kazandzidis, <a href="http://www.hms.gr/apothema/?s=sap&amp;i=20">On a conjecture of Moessner and a general problem</a>, Bull. Soc. Math. Grèce (N.S.) 2 (1961), 23-30.
%H A125714 Dexter Kozen and Alexandra Silva, <a href="https://www.jstor.org/stable/10.4169/amer.math.monthly.120.02.131">On Moessner's theorem</a>, Amer. Math. Monthly 120(2) (2013), 131-139.
%H A125714 R. Krebbers, L. Parlant, and A. Silva, <a href="https://doi.org/10.1007/978-3-319-30734-3_21">Moessner's theorem: an exercise in coinductive reasoning in Coq</a>,  Theory and practice of formal methods, 309-324, Lecture Notes in Comput. Sci., 9660, Springer, 2016.
%H A125714 Calvin T. Long, <a href="https://doi.org/10.2307/3615513">Strike it out--add it up</a>, Math. Gaz. 66 (438) (1982), 273-277.
%H A125714 Alfred Moessner, <a href="https://www.zobodat.at/pdf/Sitz-Ber-Akad-Muenchen-math-Kl_1951_0029.pdf">Eine Bemerkung über die Potenzen der natürlichen Zahlen</a>, S.-B. Math.-Nat. Kl. Bayer. Akad. Wiss., 29, 1951.
%H A125714 Ivan Paasche, <a href="https://www.zobodat.at/pdf/Sitz-Ber-Akad-Muenchen-math-Kl_1952_0001-0005.pdf">Ein neuer Beweis des Moessnerschen Satzes</a> S.-B. Math.-Nat. Kl. Bayer. Akad. Wiss. 1952 (1952), 1-5 (1953). [Two years are listed at the beginning of the journal issue.]
%H A125714 Ivan Paasche, <a href="https://doi.org/10.1007/BF01900739">Beweis des Moessnerschen Satzes mittels linearer Transformationen</a>, Arch. Math. (Basel) 6 (1955), 194-199.
%H A125714 Ivan Paasche, <a href="http://www.numdam.org/article/CM_1954-1956__12__263_0.pdf">Eine Verallgemeinerung des Moessnerschen Satzes</a>, Compositio Math. 12 (1956), 263-270.
%H A125714 Hans Salié, <a href="https://www.zobodat.at/pdf/Sitz-Ber-Akad-Muenchen-math-Kl_1952_0007-0011.pdf">Bemerkung zu einem Satz von A. Moessner</a>, S.-B. Math.-Nat. Kl. Bayer. Akad. Wiss. 1952 (1952), 7-11 (1953). [Two years are listed at the beginning of the journal issue.]
%H A125714 Oskar Perron, <a href="https://www.zobodat.at/pdf/Sitz-Ber-Akad-Muenchen-math-Kl_1951_0031-0034.pdf">Beweis des Moessnerschen Satzes</a>, S.-B. Math.-Nat. Kl. Bayer. Akad. Wiss., 31-34, 1951.
%F A125714 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.
%F A125714 Given n, let j = A003056(n-1)+1 and set t = j*(j+1)/2. Then, for 0 <= i < t, if n == -i (mod t), a(n) = abs(Stirling_1(j+1,j-i)). - _N. J. A. Sloane_, Jul 27 2021
%e A125714 An "x" prefaced before each term will indicate the term following the x being circled.
%e A125714 x1 2 x3 4 5 x6 7 8 9 x10 11 12 13 14 x15 ...
%e A125714 __x2 6 x11 18 26 x35 46 58 71 x85 ...
%e A125714 _____________x6 24 x50 96 154 x225 ...
%e A125714 _________________________x24 120 x274 ...
%e A125714 ___________________________________________x120 ...
%e A125714 ...
%e A125714 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 A125714 a := proc(n) local s,m,k,i; s := array(0..n); s[0] := 1;
%p A125714 for m from 1 to n do s[m] := 0; for k from m by - 1 to 1 do
%p A125714 for i from 1 to k do s[i] := s[i] + s[i - 1] od; lprint(s[k]);
%p A125714 if k = n then return(s[n]) fi od; lprint("-") od end: # _Peter Luschny_, Jan 27 2009
%p A125714 with(combinat);
%p A125714 s:=stirling1;
%p A125714 A003056 := proc(n) floor((sqrt(1+8*n)-1)/2) ; end proc:
%p A125714 T:=n->n*(n+1)/2; # A000217
%p A125714 g:=proc(n) local i,j,t; global T,A003056;
%p A125714 j:=A003056(n-1)+1;
%p A125714 t:=T(j);
%p A125714 for i from 0 to t-1 do
%p A125714 if ((n+i) mod t) = 0 then return(abs(s(j+1,j-i))); fi;
%p A125714 od;
%p A125714 end;
%p A125714 [seq(g(n),n=1..80)]; # _N. J. A. Sloane_, Jul 27 2021
%t A125714 n = 10; A125714 = Reap[ ClearAll[s]; s[0] = 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 A125714 (PARI) T(n, k)={ my( s=vector(n)); for( m=1, n, forstep( j=m,1,-1, s[1]++; for( i=2, j, s[i] += s[i-1]));
%o A125714 k<0 && print(vecextract(s,Str(m"..1"))));
%o A125714 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 A125714 Cf. A000217, A003056, A008275, A033312, A346004-A346007, A346595.
%K A125714 nonn,tabl
%O A125714 1,2
%A A125714 _Gary W. Adamson_, Dec 01 2006
%E A125714 More terms from _Joshua Zucker_, Jun 17 2007