A121430 -id:A121430 - cp's OEIS frontend

A121412 Triangular matrix T, read by rows, where row n of T equals row (n-1) of T^(n+1) with an appended '1'.

1, 1, 1, 3, 1, 1, 18, 4, 1, 1, 170, 30, 5, 1, 1, 2220, 335, 45, 6, 1, 1, 37149, 4984, 581, 63, 7, 1, 1, 758814, 92652, 9730, 924, 84, 8, 1, 1, 18301950, 2065146, 199692, 17226, 1380, 108, 9, 1, 1, 508907970, 53636520, 4843125, 387567, 28365, 1965, 135, 10, 1, 1

Offset: 0

Paul D. Hanna, Jul 30 2006

Related to the number of subpartitions of a partition as defined in A115728; for examples involving column k of successive matrix powers, see A121430, A121431, A121432 and A121433. Essentially the same as triangle A101479, but this form best illustrates the nice properties of this triangle.

			Triangle T begins:
1;
1, 1;
3, 1, 1;
18, 4, 1, 1;
170, 30, 5, 1, 1;
2220, 335, 45, 6, 1, 1;
37149, 4984, 581, 63, 7, 1, 1;
758814, 92652, 9730, 924, 84, 8, 1, 1;
18301950, 2065146, 199692, 17226, 1380, 108, 9, 1, 1;
508907970, 53636520, 4843125, 387567, 28365, 1965, 135, 10, 1, 1;
To get row 4 of T, append '1' to row 3 of matrix power T^5:
1;
5, 1;
25, 5, 1;
170, 30, 5, 1; ...
To get row 5 of T, append '1' to row 4 of matrix power T^6:
1;
6, 1;
33, 6, 1;
233, 39, 6, 1;
2220, 335, 45, 6, 1; ...
Likewise, get row n of T by appending '1' to row (n-1) of T^(n+1).

Alois P. Heinz, Rows n = 0..45, flattened

Cf. A121416 (T^2), A121420 (T^3), columns: A121413, A121414, A121415; related tables: A121424, A121426, A121428; related subpartitions: A121430, A121431, A121432, A121433.

T[n_, k_] := Module[{A = {{1}}, B}, Do[B = Array[0&, {m, m}]; Do[Do[B[[i, j]] = If[j == i, 1, MatrixPower[A, i][[i-1, j]]], {j, 1, i}], {i, 1, m}]; A = B, {m, 1, n+1}]; A[[n+1, k+1]]];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Oct 03 2019 *)

{T(n, k) = my(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^1)[n+1, k+1])}
for(n=0,12, for(k=0,n, print1( T(n,k),", "));print(""))

G.f.: Column k of successive powers of T satisfy the amazing relation given by: 1 = Sum_{n>=0} (1-x)^(n+1) * x^(n(n+1)/2 + k*n) * Sum_{j=0..n+k} [T^(j+1)](n+k,k) * x^j.

A121424 Rectangular table, read by antidiagonals, where row n is equal to column 0 of matrix power A121412^(n+1) for n>=0.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 3, 7, 18, 1, 4, 12, 43, 170, 1, 5, 18, 76, 403, 2220, 1, 6, 25, 118, 711, 5188, 37149, 1, 7, 33, 170, 1107, 9054, 85569, 758814, 1, 8, 42, 233, 1605, 13986, 147471, 1725291, 18301950, 1, 9, 52, 308, 2220, 20171, 225363, 2938176, 41145705

Offset: 0

Paul D. Hanna, Aug 26 2006

			Table of column 0 in matrix powers of triangle H=A121412 begins:
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,...
Rearrangement of the upper half of the table forms A121430, which is
the number of subpartitions of partition [0,1,1,2,2,2,3,3,3,3,4,...]:
1, 1,2, 3,7,12, 18,43,76,118, 170,403,711,1107,1605, 2220,...

Cf. A121425 (diagonal), A121430; rows: A101483, A121418, A121421; related tables: A121426, A121428; related triangles: A121412, A121416, A121420.

{T(n,k)=local(H=Mat(1), B); for(m=1, k+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))[k+1, 1])}

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).

A121432 Number of subpartitions of partition P=[0,0,0,1,1,1,1,2,2,2,2,2,3,...], where P(n) = [(sqrt(8*n+25) - 5)/2].

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 4, 5, 11, 18, 26, 35, 45, 101, 169, 250, 345, 455, 581, 1305, 2190, 3255, 4520, 6006, 7735, 9730, 21745, 36360, 53916, 74781, 99351, 128051, 161336, 199692, 443329, 737051, 1087583, 1502270, 1989113, 2556806, 3214774, 3973212, 4843125

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 + x + x^2)*(1-x)^0 + (x^3 + 2*x^4 + 3*x^5 + 4*x^6)*(1-x)^1 +
(5*x^7 + 11*x^8 + 18*x^9 + 26*x^10 + 35*x^11)*(1-x)^2 +
(45*x^12 + 101*x^13 + 169*x^14 + 250*x^15 + 345*x^16 + 455*x^17)*(1-x)^3 +
(581*x^18 + 1305*x^19 + 2190*x^20 + 3255*x^21 + 4520*x^22 + 6006*x^23 + 7735*x^24)*(1-x)^4 +...
When the sequence is put in the form of a triangle:
1, 1, 1,
1, 2, 3, 4,
5, 11, 18, 26, 35,
45, 101, 169, 250, 345, 455,
581, 1305, 2190, 3255, 4520, 6006, 7735,
9730, 21745, 36360, 53916, 74781, 99351, 128051, 161336, ...
then the columns of this triangle form column 2 (with offset)
of successive matrix powers of triangle H=A121412.
This sequence is embedded in table A121428 as follows.
Column 2 of successive powers of matrix H begin:
H^1: [1,1,5,45,581,9730,199692,4843125,135345925,...];
H^2: [1,2,11,101,1305,21745,443329,10679494,296547736,...];
H^3: [1,3,18,169,2190,36360,737051,17645187,487025244,...];
H^4: 1, [4,26,250,3255,53916,1087583,25889969,710546530,...];
H^5: 1,5, [35,345,4520,74781,1502270,35578270,971255050,...];
H^6: 1,6,45, [455,6006,99351,1989113,46890210,1273698270,...];
H^7: 1,7,56,581, [7735,128051,2556806,60022670,1622857887,...];
H^8: 1,8,68,724,9730, [161336,3214774,75190410,2024181693,...];
H^9: 1,9,81,885,12015,199692, [3973212,92627235,2483617140,...];
the terms enclosed in brackets form this sequence.

Cf. A121412 (triangle H), A121416 (H^2), A121420 (H^3); A121428, A121429; column 1 of H^n: A121414, A121418, A121422; variants: A121430, A121431, 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+25)+1)\2 - 2 ) )); polcoeff(A, n))}

G.f.: 1/(1-x) = Sum_{n>=0} a(n)*x^n*(1-x)^P(n), where P(n)=[(sqrt(8*n+25)-5)/2].

A121433 Number of subpartitions of partition P=[0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,...], where P(n) = [(sqrt(8*n+49) - 7)/2].

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 13, 21, 30, 40, 51, 63, 139, 229, 334, 455, 593, 749, 924, 2043, 3378, 4951, 6785, 8904, 11333, 14098, 17226, 37971, 62655, 91728, 125671, 164997, 210252, 262016, 320904, 387567, 850260, 1397268, 2038545, 2784850, 3647788

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 + x + x^2 + x^3)*(1-x)^0 +
(x^4 + 2*x^5 + 3*x^6 + 4*x^7 + 5*x^8)*(1-x)^1 +
(6*x^9 + 13*x^10 + 21*x^11 + 30*x^12 + 40*x^13 + 51*x^14)*(1-x)^2 +
(63*x^15 + 139*x^16 + 229*x^17 + 334*x^18 + 455*x^19 + 593*x^20 + 749*x^21)*(1-x)^3 +
When the sequence is put in the form of a triangle:
1, 1, 1, 1,
1, 2, 3, 4, 5,
6, 13, 21, 30, 40, 51,
63, 139, 229, 334, 455, 593, 749,
924, 2043, 3378, 4951, 6785, 8904, 11333, 14098,
17226, 37971, 62655, 91728, 125671, 164997, 210252, 262016, 320904,
then the columns of this triangle form column 3 (with offset)
of successive matrix powers of triangle H=A121412.
Column 3 of successive powers of matrix H begin:
H^1: [1,1,6,63,924,17226,387567,10182744,305379129,...];
H^2: [1,2,13,139,2043,37971,850260,22224723,663173878,...];
H^3: [1,3,21,229,3378,62655,1397268,36351147,1079567193,...];
H^4: [1,4,30,334,4951,91728,2038545,52807195,1561301733,...];
H^5: 1, [5,40,455,6785,125671,2784850,71859275,2115718545,...];
H^6: 1,6, [51,593,8904,164997,3647788,93796335,2750797677,...];
H^7: 1,7,63, [749,11333,210252,4639852,118931226,3475200792,...];
H^8: 1,8,76,924, [14098,262016,5774466,147602118,4298315847,...];
H^9: 1,9,90,1119,17226, [320904,7066029,180173970,5230303902,...];
the terms enclosed in brackets form this sequence.

Cf. A121412 (triangle H), A121416 (H^2), A121420 (H^3); column 1 of H^n: A121414, A121418, A121422; variants: A121430, A121431, A121432.

{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+49)+1)\2 - 3 ) )); polcoeff(A, n))}

G.f.: 1/(1-x) = Sum_{n>=0} a(n)*x^n*(1-x)^P(n), where P(n)=[(sqrt(8*n+49)-7)/2].

cp's OEIS Frontend

A121412 Triangular matrix T, read by rows, where row n of T equals row (n-1) of T^(n+1) with an appended '1'.

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A121424 Rectangular table, read by antidiagonals, where row n is equal to column 0 of matrix power A121412^(n+1) for n>=0.

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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).

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A121432 Number of subpartitions of partition P=[0,0,0,1,1,1,1,2,2,2,2,2,3,...], where P(n) = [(sqrt(8*n+25) - 5)/2].

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A121433 Number of subpartitions of partition P=[0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,...], where P(n) = [(sqrt(8*n+49) - 7)/2].

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