A121412 -id:A121412 - cp's OEIS frontend

A121417 Column 1 of triangle A121416.

1, 2, 9, 69, 769, 11346, 208914, 4613976, 118840164, 3496297632, 115638728395, 4246267163601, 171369282105510, 7538270885559264, 358926669220446804, 18389706733665138450, 1008742283718489346668, 58981158542987625464424

Offset: 0

Paul D. Hanna, Aug 22 2006

nonn

Also column 1 of square array A136737.

A121416 is the matrix square of triangle A121412; row n of triangle T=A121412 equals row (n-1) of T^(n+1) with an appended '1'.

Cf. A121416 (triangle); other columns: A121418, A121419.

Cf. A136737; A121413, A121422; A121427, A136741.

{a(n)=local(A=Mat(1), B); for(m=1, n+2, B=matrix(m, m); for(i=1, m, for(j=1, i, if(j==i, B[i,j]=1, B[i, j]=(A^i)[i-1, j]); )); A=B); return((A^2)[n+2, 2])}

Edited by N. J. A. Sloane, Oct 30 2008 at the suggestion of R. J. Mathar

A121421 Column 0 of triangle A121420.

Original entry on oeis.org

1, 3, 12, 76, 711, 9054, 147471, 2938176, 69328365, 1891371807, 58575539361, 2030011517685, 77827890696820, 3270046577551695, 149407542447596319, 7374639622066056408, 391044078030333899385, 22168014954558449549349

Offset: 0

Paul D. Hanna, Aug 23 2006

nonn

Also column 2 of square array A136733.

A121420 is the matrix cube of triangle A121412; row n of triangle T=A121412 equals row (n-1) of T^(n+1) with an appended '1'.

Cf. A121420 (triangle); other columns: A121422, A121423.

Cf. A136733; A101483, A121418; A121425.

{a(n)=local(A=Mat(1), B); for(m=1, n+1, B=matrix(m, m); for(i=1, m, for(j=1, i, if(j==i, B[i,j]=1, B[i, j]=(A^i)[i-1, j]); )); A=B); return((A^3)[n+1, 1])}

Edited by N. J. A. Sloane, Oct 30 2008 at the suggestion of R. J. Mathar

A121425 Main diagonal of rectangular table A121424.

Original entry on oeis.org

1, 2, 12, 118, 1605, 27816, 585046, 14459138, 410368743, 13146830110, 469123986529, 18447791712945, 792514583941223, 36925394368325295, 1854525584914459755, 99872579714406393286, 5740977285851988017769

Offset: 0

Paul D. Hanna, Aug 26 2006

nonn

Also main diagonal of square array A136733.

Cf. A121424, A121412.

Cf. A136733; A101483, A121418, A121421.

{a(n)=local(H=Mat(1), B); for(m=1, n+1, B=matrix(m, m); for(i=1, m, for(j=1, i, if(j==i, B[i, j]=1, B[i, j]=(H^i)[i-1, j]); )); H=B); return((H^(n+1))[n+1, 1])}

a(n) = [A121412^(n+1)](n,0) for n>=0; i.e., (n+1)-th term of column 0 in matrix power A121412^(n+1).

Edited by N. J. A. Sloane, Oct 30 2008 at the suggestion of R. J. Mathar

A121427 Main diagonal of rectangular table A121426.

Original entry on oeis.org

1, 2, 15, 178, 2820, 55410, 1294776, 35003430, 1073540871, 36805249870, 1394346324624, 57831360118800, 2605921998840420, 126757491839620950, 6619466939158637640, 369368127676399990338, 21932876159270004129285

Offset: 0

Paul D. Hanna, Aug 26 2006

nonn

Also main diagonal of square array A136737.

Cf. A121426, A121412.

Cf. A136737; A121413, A121417, A121422; A136741.

{a(n)=local(H=Mat(1), B); for(m=1, n+2, B=matrix(m, m); for(i=1, m, for(j=1, i, if(j==i, B[i, j]=1, B[i, j]=(H^i)[i-1, j]); )); H=B); return((H^(n+1))[n+2, 2])}

a(n) = [A121412^(n+1)](n+1,1) for n>=0; i.e., (n+1)-th term of column 1 in matrix power A121412^(n+1).

Edited by N. J. A. Sloane, Oct 30 2008 at the suggestion of R. J. Mathar

A121334 Triangle, read by rows, where T(n,k) = C( n*(n+1)/2 + n-k, n-k), for n>=k>=0.

Original entry on oeis.org

1, 2, 1, 10, 4, 1, 84, 28, 7, 1, 1001, 286, 66, 11, 1, 15504, 3876, 816, 136, 16, 1, 296010, 65780, 12650, 2024, 253, 22, 1, 6724520, 1344904, 237336, 35960, 4495, 435, 29, 1, 177232627, 32224114, 5245786, 749398, 91390, 9139, 703, 37, 1, 5317936260

Offset: 0

Paul D. Hanna, Aug 29 2006

A triangle having similar properties and complementary construction is the dual triangle A122175.

			Triangle begins:
1;
2, 1;
10, 4, 1;
84, 28, 7, 1;
1001, 286, 66, 11, 1;
15504, 3876, 816, 136, 16, 1;
296010, 65780, 12650, 2024, 253, 22, 1;
6724520, 1344904, 237336, 35960, 4495, 435, 29, 1;
177232627, 32224114, 5245786, 749398, 91390, 9139, 703, 37, 1; ...

Cf. A121439 (matrix inverse); A121412; variants: A122178, A121335, A121336; A122175 (dual).

PARI
```
T(n,k)=binomial(n*(n+1)/2+n-k,n-k)
```

Remarkably, row n of the matrix inverse (A121439) equals row n of A121412^(-n*(n+1)/2-1). Further, the following matrix products of triangles of binomial coefficients are equal: A121412 = A121334*A122178^-1 = A121335*A121334^-1 = A121336*A121335^-1, where row n of H=A121412 equals row (n-1) of H^(n+1) with an appended '1'.

A121335 Triangle, read by rows, where T(n,k) = C( n*(n+1)/2 + n-k + 1, n-k), for n>=k>=0.

Original entry on oeis.org

1, 3, 1, 15, 5, 1, 120, 36, 8, 1, 1365, 364, 78, 12, 1, 20349, 4845, 969, 153, 17, 1, 376740, 80730, 14950, 2300, 276, 23, 1, 8347680, 1623160, 278256, 40920, 4960, 465, 30, 1, 215553195, 38320568, 6096454, 850668, 101270, 9880, 741, 38, 1, 6358402050

Offset: 0

Paul D. Hanna, Aug 29 2006

A triangle having similar properties and complementary construction is the dual triangle A122176.

			Triangle begins:
1;
3, 1;
15, 5, 1;
120, 36, 8, 1;
1365, 364, 78, 12, 1;
20349, 4845, 969, 153, 17, 1;
376740, 80730, 14950, 2300, 276, 23, 1;
8347680, 1623160, 278256, 40920, 4960, 465, 30, 1;
215553195, 38320568, 6096454, 850668, 101270, 9880, 741, 38, 1; ...

Cf. A121440 (matrix inverse); A121412; variants: A122178, A121334, A121336; A122176 (dual).

PARI
```
T(n,k)=binomial(n*(n+1)/2+n-k+1,n-k)
```

Remarkably, row n of the matrix inverse (A121440) equals row n of A121412^(-n*(n+1)/2-2). Further, the following matrix products of triangles of binomial coefficients are equal: A121412 = A121334*A122178^-1 = A121335*A121334^-1 = A121336*A121335^-1, where row n of H=A121412 equals row (n-1) of H^(n+1) with an appended '1'.

A121336 Triangle, read by rows, where T(n,k) = C( n*(n+1)/2 + n-k + 2, n-k), for n>=k>=0.

Original entry on oeis.org

1, 4, 1, 21, 6, 1, 165, 45, 9, 1, 1820, 455, 91, 13, 1, 26334, 5985, 1140, 171, 18, 1, 475020, 98280, 17550, 2600, 300, 24, 1, 10295472, 1947792, 324632, 46376, 5456, 496, 31, 1, 260932815, 45379620, 7059052, 962598, 111930, 10660, 780, 39, 1

Offset: 0

Paul D. Hanna, Aug 29 2006

A triangle having similar properties and complementary construction is the dual triangle A122177.

			Triangle begins:
1;
4, 1;
21, 6, 1;
165, 45, 9, 1;
1820, 455, 91, 13, 1;
26334, 5985, 1140, 171, 18, 1;
475020, 98280, 17550, 2600, 300, 24, 1;
10295472, 1947792, 324632, 46376, 5456, 496, 31, 1;
260932815, 45379620, 7059052, 962598, 111930, 10660, 780, 39, 1; ...

Cf. A121441 (matrix inverse); A121412; variants: A122178, A121334, A121335; A122177 (dual).

PARI
```
T(n,k)=binomial(n*(n+1)/2+n-k+2,n-k)
```

Remarkably, row n of the matrix inverse (A121441) equals row n of A121412^(-n*(n+1)/2-3). Further, the following matrix products of triangles of binomial coefficients are equal: A121412 = A121334*A122178^-1 = A121335*A121334^-1 = A121336*A121335^-1, where row n of H=A121412 equals row (n-1) of H^(n+1) with an appended '1'.

A121430 Number of subpartitions of partition P=[0,1,1,2,2,2,3,3,3,3,4,...] (A003056).

Original entry on oeis.org

1, 1, 2, 3, 7, 12, 18, 43, 76, 118, 170, 403, 711, 1107, 1605, 2220, 5188, 9054, 13986, 20171, 27816, 37149, 85569, 147471, 225363, 322075, 440785, 585046, 758814, 1725291, 2938176, 4441557, 6285390, 8526057, 11226958, 14459138, 18301950

Offset: 0

Paul D. Hanna, Jul 30 2006

nonn

See A115728 for the definition of subpartitions of a partition.

			The g.f. is illustrated by:
1 = (1)*(1-x)^1 + (x + 2*x^2)*(1-x)^2 +
(3*x^3 + 7*x^5 + 12*x^6)*(1-x)^3 +
(18*x^6 + 43*x^7 + 76*x^8 + 118*x^9)*(1-x)^4 +
(170*x^10 + 403*x^11 + 711*x^12 + 1107*x^13 + 1605*x^14)*(1-x)^5 + ...
When the sequence is put in the form of a triangle:
1;
1, 2;
3, 7, 12;
18, 43, 76, 118;
170, 403, 711, 1107, 1605;
2220, 5188, 9054, 13986, 20171, 27816;
37149, 85569, 147471, 225363, 322075, 440785, 585046; ...
then the columns of this triangle form column 0 (with offset)
of successive matrix powers of triangle H=A121412.
This sequence is embedded in table A121424 as follows.
Column 0 of successive powers of matrix H begin:
H^1: [1,1,3,18,170,2220,37149,758814,18301950,...];
H^2: 1, [2,7,43,403,5188,85569,1725291,41145705,...];
H^3: 1,3, [12,76,711,9054,147471,2938176,69328365,...];
H^4: 1,4,18, [118,1107,13986,225363,4441557,103755660,...];
H^5: 1,5,25,170, [1605,20171,322075,6285390,145453290,...];
H^6: 1,6,33,233,2220, [27816,440785,8526057,195579123,...];
H^7: 1,7,42,308,2968,37149, [585046,11226958,255436293,...];
H^8: 1,8,52,396,3866,48420,758814, [14459138,326487241,...];
H^9: 1,9,63,498,4932,61902,966477,18301950, [410368743,...];
the terms enclosed in brackets form this sequence.

Cf. A121412 (triangle H), A121416 (H^2), A121420 (H^3); A121424, A121425; column 0 of H^n: A121413, A121417, A121421.

{a(n)=local(A); if(n==0,1,A=x+x*O(x^n); for(k=0, n, A+=polcoeff(A, k)*x^k*(1-(1-x)^( (sqrtint(8*k+1)+1)\2 ) )); polcoeff(A, n))}

G.f.: 1 = Sum_{n>=1} (1-x)^n * Sum_{k=n*(n-1)/2..n*(n+1)/2-1} a(k)*x^k.

A121431 Number of subpartitions of partition P=[0,0,1,1,1,2,2,2,2,3,3,3,3,3,4,...] (A052146).

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 9, 15, 22, 30, 69, 118, 178, 250, 335, 769, 1317, 1995, 2820, 3810, 4984, 11346, 19311, 29126, 41061, 55410, 72492, 92652, 208914, 352636, 528097, 740035, 993678, 1294776, 1649634, 2065146, 4613976, 7722840, 11476963, 15971180

Offset: 0

Paul D. Hanna, Jul 30 2006

nonn

See A115728 for the definition of subpartitions of a partition.

			The g.f. may be illustrated by:
1/(1-x) = (1 + 1*x)*(1-x)^0 + (x^2 + 2*x^3 + 3*x^4)*(1-x)^1 +
(4*x^5 + 9*x^6 + 15*x^7 + 22*x^8)*(1-x)^2 +
(30*x^9 + 69*x^10 + 118*x^11 + 178*x^12 + 250*x^13)*(1-x)^3 +
(335*x^14 + 769*x^15 + 1317*x^16 + 1995*x^17 + 2820*x^18 + 3810*x^19)*(1-x)^4 +...
When the sequence is put in the form of a triangle:
1, 1,
1, 2, 3,
4, 9, 15, 22,
30, 69, 118, 178, 250,
335, 769, 1317, 1995, 2820, 3810,
4984, 11346, 19311, 29126, 41061, 55410, 72492,
92652, 208914, 352636, 528097, 740035, 993678, 1294776, ...
then the columns of this triangle form column 1 (with offset)
of successive matrix powers of triangle H=A121412.
This sequence is embedded in table A121426 as follows.
Column 1 of successive powers of matrix H begin:
H^1: [1,1,4,30,335,4984,92652,2065146,53636520,...];
H^2: [1,2,9,69,769,11346,208914,4613976,118840164,...];
H^3: 1, [3,15,118,1317,19311,352636,7722840,197354133,...];
H^4: 1,4, [22,178,1995,29126,528097,11476963,291124693,...];
H^5: 1,5,30, [250,2820,41061,740035,15971180,402319275,...];
H^6: 1,6,39,335, [3810,55410,993678,21310710,533345745,...];
H^7: 1,7,49,434,4984, [72492,1294776,27611970,686872893,...];
H^8: 1,8,60,548,6362,92652, [1649634,35003430,865852191,...];
H^9: 1,9,72,678,7965,116262,2065146, [43626510,1073540871,...];
the terms enclosed in brackets form this sequence.

Cf. A121412 (triangle H), A121416 (H^2), A121420 (H^3); A121426, A121427; column 1 of H^n: A121414, A121418, A121422; variants: A121430, A121432, A121433.

{a(n)=local(A); if(n==0,1,A=x+x*O(x^n); for(k=0, n, A+=polcoeff(A, k)*x^k*(1-(1-x)^( (sqrtint(8*k+9)+1)\2 - 1 ) )); polcoeff(A, n))}

G.f.: 1/(1-x) = Sum_{n>=0} a(n)*x^n*(1-x)^A052146(n).

A122178 Triangle, read by rows, where T(n,k) = C( n*(n+1)/2 + n-k - 1, n-k), for n>=k>=0.

Original entry on oeis.org

1, 1, 1, 6, 3, 1, 56, 21, 6, 1, 715, 220, 55, 10, 1, 11628, 3060, 680, 120, 15, 1, 230230, 53130, 10626, 1771, 231, 21, 1, 5379616, 1107568, 201376, 31465, 4060, 406, 28, 1, 145008513, 26978328, 4496388, 658008, 82251, 8436, 666, 36, 1, 4431613550

Offset: 0

Paul D. Hanna, Aug 29 2006

A triangle having similar properties and complementary construction is the dual triangle A098568.

			Triangle begins:
1;
1, 1;
6, 3, 1;
56, 21, 6, 1;
715, 220, 55, 10, 1;
11628, 3060, 680, 120, 15, 1;
230230, 53130, 10626, 1771, 231, 21, 1;
5379616, 1107568, 201376, 31465, 4060, 406, 28, 1;
145008513, 26978328, 4496388, 658008, 82251, 8436, 666, 36, 1; ...

Cf. A121438 (matrix inverse); A121412; variants: A121334, A121335, A121336; A098568 (dual).

PARI
```
T(n,k)=binomial(n*(n+1)/2+n-k-1,n-k)
```

Remarkably, row n of the matrix inverse (A121438) equals row n of A121412^(-n*(n+1)/2). Further, the following matrix products of triangles of binomial coefficients are equal: A121412 = A121334*A122178^-1 = A121335*A121334^-1 = A121336*A121335^-1, where row n of H=A121412 equals row (n-1) of H^(n+1) with an appended '1'.

cp's OEIS Frontend

A121417 Column 1 of triangle A121416.

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A121421 Column 0 of triangle A121420.

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A121425 Main diagonal of rectangular table A121424.

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A121427 Main diagonal of rectangular table A121426.

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A121334 Triangle, read by rows, where T(n,k) = C( n*(n+1)/2 + n-k, n-k), for n>=k>=0.

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A121335 Triangle, read by rows, where T(n,k) = C( n*(n+1)/2 + n-k + 1, n-k), for n>=k>=0.

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A121336 Triangle, read by rows, where T(n,k) = C( n*(n+1)/2 + n-k + 2, n-k), for n>=k>=0.

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A121430 Number of subpartitions of partition P=[0,1,1,2,2,2,3,3,3,3,4,...] (A003056).

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A121431 Number of subpartitions of partition P=[0,0,1,1,1,2,2,2,2,3,3,3,3,3,4,...] (A052146).