A135876 -id:A135876 - 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.

1, 1, 1, 4, 2, 1, 28, 10, 3, 1, 280, 80, 18, 4, 1, 3640, 880, 162, 28, 5, 1, 58240, 12320, 1944, 280, 39, 6, 1, 1106560, 209440, 29160, 3640, 418, 52, 7, 1, 24344320, 4188800, 524880, 58240, 5714, 600, 66, 8, 1, 608608000, 96342400, 11022480, 1106560, 95064

Offset: 0

Paul D. Hanna, Dec 22 2007

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.

			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),19,20,21,..;
(4),(10),(18),28,(39),52,(66),82,(99),118,138,(159),182,206,(231),258,286,..;
(28),(80),(162),280,(418),600,(806),1064,(1350),1696,2074,(2485),2966,3484,..;
(280),(880),(1944),3640,(5714),8680,(12164),16840,(22194),29080,36824,(45474),.;
(3640),(12320),(29160),58240,(95064),151200,(219108),315440,(428652),581680,...;
(58240),(209440),(524880),1106560,(1864456),3082240,...;
where terms in parenthesis are at positions {[m*(m+5)/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 {[m*(m+5)/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 [0,1,2,4,6,8,11,14,17,...] to get:
[4, 6, 8, 10,11, 13,14, 16,17, 19,20,21, 23,24,25, 27,28,29, ...]
then take partial sums to obtain row 2:
[4, 10, 18, 28,39, 52,66, 82,99, 118,138,159, 182,206,231, ...].
Continuing in this way will generate all the rows of this array.

Cf. columns: A007559, A008544, A032031, A024216, A024395; related tables: A136213, A125714, A135876, A127054, A125781; A136217.

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 *)

{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}

Columns 0, 1 and 2 form the triple factorials A007559, A008544 and A032031, respectively. Column 4 equals A024216; column 6 equals A024395.

A135878 Square array, read by antidiagonals, where row n+1 is generated from row n by first removing terms at positions [(m+3)^2/4 - 2] for m>=0 and then taking partial sums, starting with all 1's in row 0.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 6, 6, 3, 1, 25, 25, 12, 4, 1, 138, 138, 63, 19, 5, 1, 970, 970, 421, 113, 28, 6, 1, 8390, 8390, 3472, 832, 190, 38, 7, 1, 86796, 86796, 34380, 7420, 1560, 283, 50, 8, 1, 1049546, 1049546, 399463, 78406, 15250, 2502, 411, 63, 9, 1, 14563135, 14563135

Offset: 0

Paul D. Hanna, Dec 14 2007

Column 0 is A135881 which equals column 0 of triangle A135879 and also equals column 0 of triangle A135880. Compare to triangle A135879, which is generated by a complementary process. An interesting variant is square array A135876, in which column 0 equals the double factorials (A001147).

			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,...;
(2),6,(12),19,(28),38,50,(63),77,93,(110),128,148,169,(191),214,...;
(6),25,(63),113,(190),283,411,(559),728,942,(1181),1446,1766,2116,...;
(25),138,(421),832,(1560),2502,3948,(5714),7830,10740,(14130),18036,...;
(138),970,(3472),7420,(15250),25990,44026,(67112),95918,138343,(189598),..;
(970),8390,(34380),78406,(174324),312667,(563287),897471,1329234,2003240,..;
(8390),86796,(399463),962750,(2291984),4295224,8168819,(13523882),20656067,.;
(86796),1049546,(5344770),13513589,(34169656),66534382,132787852,(227380975),.;
(1049546),14563135,(81097517),213885369,(570682050),1149537869,2395865161,..;
(14563135),228448504,(1377986373),3773851534,(10568874312),21945438536,...;
where terms in parenthesis are removed before taking partial sums.
For example, to generate row 2 from row 1, remove terms at positions
{[(m+3)^2/4-2], m>=0} = [0,2,4,7,10,14,18,23,28,34,...] to obtain:
[2, 4, 6,7, 9,10, 12,13,14, 16,17,18, 20,21,22,23, ...]
then take partial sums to get row 2:
[2, 6, 12,19, 28,38, 50,63,77, 93,110,128, 148,169,191,214, ...].
Repeating this process will generate all the rows of the triangle.
Triangle A135880 begins:
1;
1, 1;
2, 2, 1;
6, 7, 3, 1;
25, 34, 15, 4, 1;
138, 215, 99, 26, 5, 1;
970, 1698, 814, 216, 40, 6, 1; ...
and is generated by matrix powers of itself.

Cf. A135881, A135879, A135880; variants: A135876, A125714.

{T(n, k)=local(A=0, b=0, c=0, d=0); if(n==0, A=1, until(d>k, if(c==floor((b+3)^2/4)-2, b+=1, A+=T(n-1, c); d+=1); c+=1)); A}

A152405 Square array, read by antidiagonals, where row n+1 is generated from row n by first removing terms in row n at positions {m*(m+1)/2, m>=0} and then taking partial sums, starting with all 1's in row 0.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 14, 8, 3, 1, 86, 45, 14, 4, 1, 645, 318, 86, 22, 5, 1, 5662, 2671, 645, 152, 31, 6, 1, 56632, 25805, 5662, 1251, 232, 41, 7, 1, 633545, 280609, 56632, 11869, 2026, 327, 53, 8, 1, 7820115, 3381993, 633545, 126987, 20143, 2991, 457, 66, 9, 1

Offset: 0

Paul D. Hanna, Dec 05 2008

			Table begins:
(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),...;
(3),(8),14,(22),31,41,(53),66,80,95,(112),130,149,169,190,...;
(14),(45),86,(152),232,327,(457),606,775,965,(1202),1464,1752,2067,...;
(86),(318),645,(1251),2026,2991,(4455),6207,8274,10684,(13934),17653,...;
(645),(2671),5662,(11869),20143,30827,(48480),70355,96990,128959,...;
(5662),(25805),56632,(126987),223977,352936,(582183),874664,1240239,...;
(56632),(280609),633545,(1508209),2748448,4438122,(7641111),11831184,...;
(633545),(3381993),7820115,(19651299),36837937,60743909,...; ...
where row n equals the partial sums of row n-1 after removing terms
at positions {m*(m+1)/2, m>=0} (marked by parenthesis in above table).
For example, to generate row 3 from row 2:
[3,8, 14, 22, 31,41, 53, 66,80,95, 112, 130,...]
remove terms at positions {0,1,3,6,10,...}, yielding:
[14, 31,41, 66,80,95, 130,149,169,190, ...]
then take partial sums to obtain row 3:
[14, 45,86, 152,232,327, 457,606,775,965, ...].
Continuing in this way generates all rows of this table.
RELATION TO POWERS OF A SPECIAL TRIANGULAR MATRIX.
Columns 0 and 1 are found in triangle T=A152400, which begins:
1;
1, 1;
3, 2, 1;
14, 8, 3, 1;
86, 45, 15, 4, 1;
645, 318, 99, 24, 5, 1;
5662, 2671, 794, 182, 35, 6, 1;
56632, 25805, 7414, 1636, 300, 48, 7, 1; ...
where column k of T = column 0 of matrix power T^(k+1) for k>=0.
Furthermore, matrix powers of triangle T=A152400 satisfy:
column k of T^(j+1) = column j of T^(k+1) for all j>=0, k>=0.
Column 3 of this square array = column 1 of T^2:
1;
2, 1;
8, 4, 1;
45, 22, 6, 1;
318, 152, 42, 8, 1;
2671, 1251, 345, 68, 10, 1;
25805, 11869, 3253, 648, 100, 12, 1; ...
RELATED TRIANGLE A127714 begins:
1;
1, 1, 1;
1, 2, 2, 3, 3, 3;
1, 3, 5, 5, 8, 11, 11, 14, 14, 14;
1, 4, 9, 14, 14, 22, 33, 44, 44, 58, 72, 72, 86, 86, 86;...
where right border = column 0 of this square array.

Cf. columns: A127715, A152401, A152404.
Cf. related triangles: A152400, A127714.
Cf. variants: A125781, A127054, A136212, A136217, A135876, A135878, A136730.

{T(n, k)=local(A=0, m=0, c=0, d=0); if(n==0, A=1, until(d>k, if(c==m*(m+1)/2, m+=1, A+=T(n-1, c); d+=1); c+=1)); A}

A135877 Triangle, read by rows of n(n+1)+1 terms, where row n+1 is generated from row n by inserting zeros at positions [(m+2)^2/4 - 1] for m=1..2n+2 and then taking partial sums from right to left, starting with a single 1 in row 0.

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 3, 2, 2, 1, 1, 15, 15, 15, 12, 12, 9, 9, 6, 4, 4, 2, 1, 1, 105, 105, 105, 90, 90, 75, 75, 60, 48, 48, 36, 27, 27, 18, 12, 8, 8, 4, 2, 1, 1, 945, 945, 945, 840, 840, 735, 735, 630, 540, 540, 450, 375, 375, 300, 240, 192, 192, 144, 108, 81, 81, 54, 36, 24, 16, 16, 8

Offset: 0

Paul D. Hanna, Dec 14 2007

Compare to square array A135876 which is generated by a complementary process. Compare to triangle A127452 which generates the factorials in column 0. A very interesting variant is triangle A135879.

			Triangle begins:
1;
1, 1, 1;
3, 3, 3, 2, 2, 1, 1;
15, 15, 15, 12, 12, 9, 9, 6, 4, 4, 2, 1, 1;
105, 105, 105, 90, 90, 75, 75, 60, 48, 48, 36, 27, 27, 18, 12, 8, 8, 4, 2, 1, 1;
945, 945, 945, 840, 840, 735, 735, 630, 540, 540, 450, 375, 375, 300, 240, 192, 192, 144, 108, 81, 81, 54, 36, 24, 16, 16, 8, 4, 2, 1, 1; ...
To generate the triangle, start with a single 1 in row 0,
and then obtain row n+1 from row n by inserting zeros
at positions [(m+2)^2/4 - 1] for m=1..2n+2 and then
taking reverse partial sums (i.e., summing from right to left).
Start with row 0, insert 2 zeros in front of the '1':
[0,0,1];
take reverse partial sums to get row 1:
[1,1,1];
insert zeros at positions [0,1,3,5]:
[0,0,1,0,1,0,1];
take reverse partial sums to get row 2:
[3,3,3,2,2,1,1];
insert zeros at positions [0,1,3,5,8,11]:
[0,0,3,0,3,0,3,2,0,2,1,0,1];
take reverse partial sums to get row 3:
[15,15,15,12,12,9,9,6,4,4,2,1,1];
insert zeros at positions [0,1,3,5,8,11,15,19]:
[0,0,15,0,15,0,15,12,0,12,9,0,9,6,4,0,4,2,1,0,1];
take reverse partial sums to get row 4:
[105,105,105,90,90,75,75,60,48,48,36,27,27,18,12,8,8,4,2,1,1].

Cf. A001147; A135876; variants: A135879, A127452, A125781.

{T(n,k)=local(A=[1],B);if(n>0,for(i=1,n,m=1;B=[0]; for(j=1,#A,if(j+m-1==floor((m+2)^2/4)-1,m+=1;B=concat(B,0));B=concat(B,A[j])); A=Vec(Polrev(Vec(Pol(B)/(1-x+O(x^#B)))))));if(k+1>#A,0,A[k+1])} /* for(n=0,8,for(k=0,n*(n+1),print1(T(n,k),","));print("")) */

Column 0 equals the double factorials A001147(n) = (2n)!/(n!*2^n).

A156628 Square array, read by antidiagonals, where row n+1 is generated from row n by first removing terms in row n at positions 0 and {(m+1)*(m+2)/2-2, m>0} and then taking partial sums, starting with all 1's in row 0.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 13, 7, 3, 1, 71, 33, 13, 4, 1, 461, 191, 71, 20, 5, 1, 3447, 1297, 461, 120, 28, 6, 1, 29093, 10063, 3447, 836, 181, 38, 7, 1, 273343, 87669, 29093, 6616, 1333, 270, 49, 8, 1, 2829325, 847015, 273343, 58576, 11029, 2150, 375, 61, 9, 1

Offset: 0

Paul D. Hanna, Feb 17 2009

			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 0 and {(m+1)*(m+2)/2-2,m>0} and then taking partial sums.
This square array A begins:
(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, ...;
(3), (7), 13, 20, (28), 38, 49, 61, (74), 89, 105, 122, 140, (159),...;
(13), (33), 71, 120, (181), 270, 375, 497, (637), 817, 1019, 1244, ...;
(71), (191), 461, 836, (1333), 2150, 3169, 4413, (5906), 8001, ...;
(461), (1297), 3447, 6616, (11029), 19030, 29483, 42775, (59324),...;
(3447), (10063), 29093, 58576, (101351), 185674, 300329, 451277, ...;
(29093), (87669), 273343, 573672, (1024949), 1982310, 3330651, ...;
(273343), (847015), 2829325, 6159976, (11320359), 23009602, 39998897, ...;
where terms in parenthesis at positions {0,1,4,8,13,..} in a row
are removed before taking partial sums to obtain the next row.
...
RELATION TO SPECIAL TRIANGLE.
Triangle A104980 begins:
1;
1, 1;
3, 2, 1;
13, 7, 3, 1;
71, 33, 13, 4, 1;
461, 191, 71, 21, 5, 1;
3447, 1297, 461, 133, 31, 6, 1;
29093, 10063, 3447, 977, 225, 43, 7, 1; ...
in which column 0 and column 1 are found in square array A.
...
Matrix square of A104980 = triangle A104988 which begins:
1;
2, 1;
8, 4, 1;
42, 20, 6, 1;
266, 120, 38, 8, 1;
1954, 836, 270, 62, 10, 1;
16270, 6616, 2150, 516, 92, 12, 1;
151218, 58576, 19030, 4688, 882, 128, 14, 1; ...
where column 1 and column 2 are also found in square array A.

Cf. columns: A003319, A104981, A156629, related triangles: A104980, A104988.

Cf. related tables: A136212, A136213, A125714, A135876, A127054, A125781, A136217.

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

Column 0 = Column 0 of triangle A104980 = A003319.
Column 1 = Column 1 of triangle A104980 = A104981.
Column 3 = column 1 of A104988 (matrix square of A104980).
Column 5 = column 2 of A104988 (matrix square of A104980).

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.

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A135878 Square array, read by antidiagonals, where row n+1 is generated from row n by first removing terms at positions [(m+3)^2/4 - 2] for m>=0 and then taking partial sums, starting with all 1's in row 0.

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A152405 Square array, read by antidiagonals, where row n+1 is generated from row n by first removing terms in row n at positions {m*(m+1)/2, m>=0} and then taking partial sums, starting with all 1's in row 0.

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A135877 Triangle, read by rows of n(n+1)+1 terms, where row n+1 is generated from row n by inserting zeros at positions [(m+2)^2/4 - 1] for m=1..2n+2 and then taking partial sums from right to left, starting with a single 1 in row 0.

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A156628 Square array, read by antidiagonals, where row n+1 is generated from row n by first removing terms in row n at positions 0 and {(m+1)*(m+2)/2-2, m>0} and then taking partial sums, starting with all 1's in row 0.

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