A125781 -id:A125781 - cp's OEIS frontend

A127054 Rectangular table, read by antidiagonals, defined by the following rule: start with all 1's in row zero; from then on, row n+1 equals the partial sums of row n excluding terms in columns k = m*(m+1)/2 (m>=1).

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 10, 9, 4, 1, 1, 34, 33, 15, 5, 1, 1, 154, 153, 65, 23, 6, 1, 1, 874, 873, 339, 119, 32, 7, 1, 1, 5914, 5913, 2103, 719, 186, 42, 8, 1, 1, 46234, 46233, 15171, 5039, 1230, 267, 54, 9, 1, 1, 409114, 409113, 124755, 40319, 9258, 1891, 380

Offset: 0

Paul D. Hanna, Jan 04 2007

Variant of table A125781. Generated by a method similar to Moessner's factorial triangle (A125714).

			Rows are partial sums excluding terms in columns k = {1,3,6,10,...}:
row 2 = partial sums of [1, 3, 5,6, 8,9,10, 12,13,14,15, ...];
row 3 = partial sums of [1, 9, 23,32, 54,67,81, 113,131,150,170, ...];
row 4 = partial sums of [1, 33, 119,186, 380,511,661, 1045,1283,...].
The terms that are excluded in the partial sums are shown enclosed in
parenthesis in the table below. Rows of this table begin:
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), ...;
1,(4), 9, (15), 23, 32, (42), 54, 67, 81, (96), 113, 131, 150, ...;
1,(10), 33, (65), 119, 186, (267), 380, 511, 661, (831), 1045, ...;
1,(34), 153, (339), 719, 1230, (1891), 2936, 4219, 5765, (7600), ...;
1,(154), 873, (2103), 5039, 9258, (15023), 25148, 38203, 54625, ..;
1,(874), 5913, (15171), 40319, 78522, (133147), 238124, 379339, ...;
1,(5914), 46233, (124755), 362879, 742218, (1305847), 2477468, ...;
1,(46234), 409113, (1151331), 3628799, 7742058, (14059423), ...;
1,(409114), 4037913, (11779971), 39916799, 88369098, (164977399),...;
Columns include:
k=1: A003422 (Left factorials: !n = Sum k!, k=0..n-1);
k=2: A007489 (Sum of k!, k=1..n);
k=3: A097422 (Sum{k=1 to n} H(k) k!, where H(k) = sum{j=1 to k} 1/j);
k=4: A033312 (n! - 1);
k=5: Partial sums of A001705;
k=6: partial sums of A000399 (Stirling numbers of first kind s(n,3)).

Cf. variants: A125781, A125714; antidiagonal sums: A127055; diagonal: A127056; columns: A003422, A007489, A097422, A033312.

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

A135879 Triangle, read by rows of A135901(n) terms, where row n+1 is generated from row n by inserting zeros at positions [(m+3)^2/4 - 2], as m=0,1,2,3,... 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, 2, 2, 1, 1, 6, 6, 4, 4, 2, 2, 1, 25, 25, 19, 19, 13, 13, 9, 5, 5, 3, 1, 1, 138, 138, 113, 113, 88, 88, 69, 50, 50, 37, 24, 24, 15, 10, 5, 5, 2, 1, 970, 970, 832, 832, 694, 694, 581, 468, 468, 380, 292, 292, 223, 173, 123, 123, 86, 62, 38, 38, 23, 13, 8, 3, 3, 1, 8390

Offset: 0

Paul D. Hanna, Dec 14 2007

Column 0 is A135881 which equals column 0 of square array A135878 and also equals column 0 of triangle A135880. Compare to square array A135878, which is generated by a complementary process. An interesting variant is triangle A135877 in which column 0 equals the double factorials (A001147).

			Triangle begins:
1;
1, 1;
2, 2, 1, 1;
6, 6, 4, 4, 2, 2, 1;
25, 25, 19, 19, 13, 13, 9, 5, 5, 3, 1, 1;
138, 138, 113, 113, 88, 88, 69, 50, 50, 37, 24, 24, 15, 10, 5, 5, 2, 1;
970, 970, 832, 832, 694, 694, 581, 468, 468, 380, 292, 292, 223, 173, 123, 123, 86, 62, 38, 38, 23, 13, 8, 3, 3, 1;
8390, 8390, 7420, 7420, 6450, 6450, 5618, 4786, 4786, 4092, 3398, 3398, 2817, 2349, 1881, 1881, 1501, 1209, 917, 917, 694, 521, 398, 275, 275, 189, 127, 89, 51, 51, 28, 15, 7, 4, 1, 1;
There are A135901(n) number of terms in row n.
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+3)^2/4 - 2], m=0,1,2,...} and then
taking reverse partial sums (i.e., summing from right to left).
Start with row 0, insert a zero in front of the '1' at position 0:
[0,1];
take reverse partial sums to get row 1:
[1,1];
insert zeros at positions [0,2]:
[0,1,0,1];
take reverse partial sums to get row 2:
[2,2,1,1];
insert zeros at positions [0,2,4]:
[0,2,0,2,0,1,1];
take reverse partial sums to get row 3:
[6,6,4,4,2,2,1];
insert zeros at positions [0,2,4,7]:
[0,6,0,6,0,4,4,0,2,2,0,1];
take reverse partial sums to get row 4:
[25,25,19,19,13,13,9,5,5,3,1,1];
insert zeros at positions [0,2,4,7,10,14]:
[0,25,0,25,0,19,19,0,13,13,0,9,5,5,0,3,1,1];
take reverse partial sums to get row 5:
[138,138,113,113,88,88,69,50,50,37,24,24,15,10,5,5,2,1].
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, A135901, A135878, A135880; variants: A135877, A127452, A125781.

{T(n,k)=local(A=[1],B);if(n>0,for(i=1,n,m=1;B=[]; 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])}

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).

A127420 Triangle, read by rows, where row n+1 is generated from row n by first inserting zeros at positions {(m+2)*(m+3)/2, m>=0} in row n and then taking the partial sums in reverse order, for n>=2, starting with 1's in the initial two rows.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 4, 2, 2, 1, 9, 5, 5, 3, 1, 1, 24, 15, 15, 10, 5, 5, 2, 1, 77, 53, 53, 38, 23, 23, 13, 8, 3, 3, 1, 295, 218, 218, 165, 112, 112, 74, 51, 28, 28, 15, 7, 4, 1, 1, 1329, 1034, 1034, 816, 598, 598, 433, 321, 209, 209, 135, 84, 56, 28, 28, 13, 6, 2, 1, 6934, 5605

Offset: 0

Paul D. Hanna, Jan 14 2007

Column 0 forms A091352, which also equals column 1 of table A125781, where table A125781 is generated by a complementary recurrence of this triangle. The number of terms in row n is A127419(n).

			To generate row 6, start with row 5:
24, 15, 15, 10, 5, 5, 2, 1;
insert zeros at positions [1,4,8,13,..., (m+2)*(m+3)/2 - 2,...]:
24, 0, 15, 15, 0, 10, 5, 5, 0, 2, 1;
then row 6 equals the partial sums of row 5 taken in reverse order:
24, _0, 15, 15, _0, 10, _5, 5, 0, 2, 1;
77, 53, 53, 38, 23, 23, 13, 8, 3, 3, 1.
Triangle begins:
1;
1, 1;
2, 1, 1;
4, 2, 2, 1;
9, 5, 5, 3, 1, 1;
24, 15, 15, 10, 5, 5, 2, 1;
77, 53, 53, 38, 23, 23, 13, 8, 3, 3, 1;
295, 218, 218, 165, 112, 112, 74, 51, 28, 28, 15, 7, 4, 1, 1;
1329, 1034, 1034, 816, 598, 598, 433, 321, 209, 209, 135, 84, 56, 28, 28, 13, 6, 2, 1;
Column 0 of this triangle equals column 1 of triangle A091351, where triangle A091351 begins:
1;
1, 1;
1, 2, 1;
1, 4, 3, 1;
1, 9, 9, 4, 1;
1, 24, 30, 16, 5, 1;
1, 77, 115, 70, 25, 6, 1;
1, 295, 510, 344, 135, 36, 7, 1; ...
and column k of A091351 = row sums of matrix power A091351^k for k>=0.

Cf. A091352, A091351, A125781, A127419.

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

A127054 Rectangular table, read by antidiagonals, defined by the following rule: start with all 1's in row zero; from then on, row n+1 equals the partial sums of row n excluding terms in columns k = m*(m+1)/2 (m>=1).

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Maple

A135879 Triangle, read by rows of A135901(n) terms, where row n+1 is generated from row n by inserting zeros at positions [(m+3)^2/4 - 2], as m=0,1,2,3,... and then taking partial sums from right to left, starting with a single 1 in row 0.

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PARI

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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PARI

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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Formula

A127420 Triangle, read by rows, where row n+1 is generated from row n by first inserting zeros at positions {(m+2)*(m+3)/2, m>=0} in row n and then taking the partial sums in reverse order, for n>=2, starting with 1's in the initial two rows.

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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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