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A135876 Square array, read by antidiagonals, where row n+1 is generated from row n by first removing terms at positions [(m+2)^2/4 - 1] for m>=0 and then taking partial sums, starting with all 1's in row 0. 5
1, 1, 1, 3, 2, 1, 15, 8, 3, 1, 105, 48, 15, 4, 1, 945, 384, 105, 23, 5, 1, 10395, 3840, 945, 176, 33, 6, 1, 135135, 46080, 10395, 1689, 279, 44, 7, 1, 2027025, 645120, 135135, 19524, 2895, 400, 57, 8, 1, 34459425, 10321920, 2027025, 264207, 35685, 4384, 561 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

This is the double factorial analog of Moessner's factorial array (A125714). Compare to triangle A135877 which is generated by a complementary process. A very interesting variant is square array A135878.

LINKS

Table of n, a(n) for n=0..51.

FORMULA

T(n,0) = (2n)!/n!/2^n; T(n,1) = 2^n*n!; T(n,2) = (2n+1)!/n!/2^n; T(n,3) = A004041(n) = (2n+1)!/n!/2^n * Sum_{k=0..n} 1/(2k+1). T(n,4) = A129890(n) = 2^(n+1)*(n+1)! - (2n+1)!/n!/2^n = T(n+1,1)-T(n+1,0).

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),...;

(1),(2),3,(4),5,(6),7,8,(9),10,11,(12),13,14,15,(16),17,18,19,(20),...;

(3),(8),15,(23),33,(44),57,71,(86),103,121,(140),161,183,206,(230),..;

(15),(48),105,(176),279,(400),561,744,(950),1206,1489,(1800),2171,..;

(105),(384),945,(1689),2895,(4384),6555,9129,(12139),16161,20763,..;

(945),(3840),10395,(19524),35685,(56448),89055,129072,(177331),245778,...;

(10395),(46080),135135,(264207),509985,(836352),1381905,2071215,(2924172),.;

(135135),(645120),2027025,(4098240),8294895,(14026752),24137505,...; ...

where terms in parenthesis are removed before taking partial sums.

For example, to generate row 2 from row 1, remove terms at positions

{[(m+2)^2/4-1], m>=0} = [0,1,3,5,8,11,15,19,24,29,35,...] to obtain:

[3, 5, 7,8, 10,11, 13,14,15, 17,18,19, 21,22,23,24, 25,26,27,28, ...]

then take partial sums to get row 2:

[3, 8, 15,23, 33,44, 57,71,86, 103,121,140, 161,183,206,230, ...].

Repeating this process will generate all the rows of the triangle,

where column 0 will be the odd double factorials (A001147)

and column 1 will be the even double factorials (A000165).

PROG

(PARI) {T(n, k)=local(A=0, b=0, c=0, d=0); if(n==0, A=1, until(d>k, if(c==floor((b+2)^2/4)-1, b+=1, A+=T(n-1, c); d+=1); c+=1)); A}

CROSSREFS

Cf. columns: A001147, A000165, A004041, A129890; variants: A135878, A125714.

Sequence in context: A140709 A109282 A135902 * A136217 A166884 A136220

Adjacent sequences:  A135873 A135874 A135875 * A135877 A135878 A135879

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Dec 14 2007

STATUS

approved

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Last modified September 26 23:42 EDT 2021. Contains 347673 sequences. (Running on oeis4.)