A136217 Square array, read by antidiagonals, where row n+1 is generated from row n by first removing terms in row n at positions {floor(m*(m+7)/6), m>=0} and then taking partial sums, starting with all 1's in row 0.

1, 1, 1, 3, 2, 1, 15, 8, 3, 1, 108, 49, 15, 4, 1, 1036, 414, 108, 24, 5, 1, 12569, 4529, 1036, 198, 34, 6, 1, 185704, 61369, 12569, 2116, 306, 46, 7, 1, 3247546, 996815, 185704, 28052, 3493, 453, 59, 8, 1, 65762269, 18931547, 3247546, 446560, 48800, 5555, 622, 74, 9, 1

Offset

0,4

Comments

A variant of the triple factorial array A136212. Compare to triangle array A136218, which is generated by a complementary process.

Links

Table of n, a(n) for n=0..54.

Formula

Let triangular matrix P = A136220, then: column 0 (A136221) = column 0 of P; column 1 (A136226) = column 0 of P^2; column 3 (A136229) = column 0 of P^4.

Example

Square array begins:
(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),(2),3,(4),5,(6),7,(8),9,10,(11),12,13,(14),15,16,(17),18,19,20,(21),..;
(3),(8),15,(24),34,(46),59,(74),90,108,(127),147,169,(192),216,242,(269),..;
(15),(49),108,(198),306,(453),622,(838),1080,1377,(1704),2062,2485,(2943),..;
(108),(414),1036,(2116),3493,(5555),8040,(11477),15483,20748,(26748),33528,..;
(1036),(4529),12569,(28052),48800,(82328),124335,(186261),260856,364551,..;
(12569),(61369),185704,(446560),811111,(1438447),2250731,(3513569),5078154,..;
(185704),(996815),3247546,(8325700),15684001,(29039188),46830722,...;
(3247546),(18931547),65762269,(178284892),346583419,...;
(65762269),(412345688),1515642725,(4317391240),...; ...
where terms in parenthesis are at positions {floor(m*(m+7)/6), m>=0} and are removed before taking partial sums to obtain the next row.
To generate the array, start with all 1's in row 0; from then on, obtain row n+1 from row n by first removing terms in row n at positions {floor(m*(m+7)/6), m>=0} and then taking partial sums.
For example, to generate row 2 from row 1:
[(1),(2),3,(4),5,(6),7,(8),9,10,(11),12,13,(14),15,16,(17),18,...],
remove terms at positions [0,1,3,5,7,10,13,16,20,...] to get:
[3, 5, 7, 9,10, 12,13, 15,16, 18,19,20, 22,23,24, 26,27,28,...]
then take partial sums to obtain row 2:
[3,8,15,24,34,46,59,74,90,108,127,147,169,192,216,242,269,...].
Continuing in this way will generate all the rows of this array.
Amazingly, column 0 of this array = column 0 of triangle P=A136220:
       1;
       1,      1;
       3,      2,     1;
      15,     10,     3,    1;
     108,     75,    21,    4,   1;
    1036,    753,   208,   36,   5,  1;
   12569,   9534,  2637,  442,  55,  6, 1;
  185704, 146353, 40731, 6742, 805, 78, 7, 1; ...
where column k of P^3 = column 0 of P^(3k+3) such that column 0 of P^3 = column 0 of P shift one place left.

Mathematica

nmax = 9;
row[0] = Table[1, {nmax^2}];
row[n_] := row[n] = Accumulate[Delete[row[n-1], Table[{Floor[m((m+7)/6)+1] }, {m, 0, (1/2)(-7 + Sqrt[1 + 24 Length[row[n-1]]]) // Floor}]]];
R = row /@ Range[0, nmax];
T[n_, k_] := R[[n+1, k+1]];
Table[T[n-k, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 06 2019 *)

Prog

(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+7))\6, m+=1, A+=T(n-1, c); d+=1); c+=1)); A}

Crossrefs

Cf. columns: A136221, A136226, A136229; related tables: A136220 (P), A136226 (P^2), A136232 (P^4).

Sequence in context: A109282 A135902 A135876 * A166884 A136220 A248035

Adjacent sequences: A136214 A136215 A136216 * A136218 A136219 A136220

Keyword

nice,nonn,tabl

Author

Paul D. Hanna, Dec 23 2007

Status

approved