cp's OEIS Frontend

A136212 Triple factorial array, read by antidiagonals, where row n+1 is generated from row n by first removing terms in row n at positions {[m*(m+5)/6], m >= 0} and then taking partial sums, starting with all 1's in row 0.

%I A136212 #12 Mar 06 2024 08:31:39
%S A136212 1,1,1,4,2,1,28,10,3,1,280,80,18,4,1,3640,880,162,28,5,1,58240,12320,
%T A136212 1944,280,39,6,1,1106560,209440,29160,3640,418,52,7,1,24344320,
%U A136212 4188800,524880,58240,5714,600,66,8,1,608608000,96342400,11022480,1106560,95064
%N A136212 Triple factorial array, read by antidiagonals, where row n+1 is generated from row n by first removing terms in row n at positions {[m*(m+5)/6], m >= 0} and then taking partial sums, starting with all 1's in row 0.
%C A136212 This is the triple factorial variant of Moessner's factorial array described by A125714 and also of the double factorial array A135876. Another very interesting variant is A136217.
%F A136212 Columns 0, 1 and 2 form the triple factorials A007559, A008544 and A032031, respectively. Column 4 equals A024216; column 6 equals A024395.
%e A136212 Square array begins:
%e A136212 (1),(1),(1),1,(1),1,(1),1,(1),1,1,(1),1,1,(1),1,1,(1),1,1,1,(1),1,1,1,...;
%e A136212 (1),(2),(3),4,(5),6,(7),8,(9),10,11,(12),13,14,(15),16,17,(18),19,20,21,..;
%e A136212 (4),(10),(18),28,(39),52,(66),82,(99),118,138,(159),182,206,(231),258,286,..;
%e A136212 (28),(80),(162),280,(418),600,(806),1064,(1350),1696,2074,(2485),2966,3484,..;
%e A136212 (280),(880),(1944),3640,(5714),8680,(12164),16840,(22194),29080,36824,(45474),.;
%e A136212 (3640),(12320),(29160),58240,(95064),151200,(219108),315440,(428652),581680,...;
%e A136212 (58240),(209440),(524880),1106560,(1864456),3082240,...;
%e A136212 where terms in parenthesis are at positions {[m*(m+5)/6], m>=0}
%e A136212 and are removed before taking partial sums to obtain the next row.
%e A136212 To generate the array, start with all 1's in row 0; from then on,
%e A136212 obtain row n+1 from row n by first removing terms in row n at
%e A136212 positions {[m*(m+5)/6], m>=0} and then taking partial sums.
%e A136212 For example, to generate row 2 from row 1:
%e A136212 [(1),(2),(3),4,(5),6,(7),8,(9),10,11,(12),13,14,(15),16,17,(18),...],
%e A136212 remove terms at positions [0,1,2,4,6,8,11,14,17,...] to get:
%e A136212 [4, 6, 8, 10,11, 13,14, 16,17, 19,20,21, 23,24,25, 27,28,29, ...]
%e A136212 then take partial sums to obtain row 2:
%e A136212 [4, 10, 18, 28,39, 52,66, 82,99, 118,138,159, 182,206,231, ...].
%e A136212 Continuing in this way will generate all the rows of this array.
%t A136212 t[n_, k_] := t[n, k] = Module[{a = 0, m = 0, c = 0, d = 0}, If[n == 0, a = 1, While[d <= k, If[c == Quotient[(m*(m + 5)), 6], m += 1, a += t[n - 1, c]; d += 1]; c += 1]]; a]; Table[t[n - k, k], {n, 0, 9}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Mar 06 2013, translated from Pari *)
%o A136212 (PARI) {T(n, k)=local(A=0, m=0, c=0, d=0); if(n==0, A=1, until(d>k, if(c==(m*(m+5))\6, m+=1, A+=T(n-1, c); d+=1); c+=1)); A}
%Y A136212 Cf. columns: A007559, A008544, A032031, A024216, A024395; related tables: A136213, A125714, A135876, A127054, A125781; A136217.
%K A136212 nice,nonn,tabl
%O A136212 0,4
%A A136212 _Paul D. Hanna_, Dec 22 2007