A121429 -id:A121429 - cp's OEIS frontend

A121428 Rectangular table, read by antidiagonals, where row n is equal to column 2 of matrix power A121412^(n+1) for n>=0.

1, 1, 1, 1, 2, 5, 1, 3, 11, 45, 1, 4, 18, 101, 581, 1, 5, 26, 169, 1305, 9730, 1, 6, 35, 250, 2190, 21745, 199692, 1, 7, 45, 345, 3255, 36360, 443329, 4843125, 1, 8, 56, 455, 4520, 53916, 737051, 10679494, 135345925, 1, 9, 68, 581, 6006, 74781, 1087583, 17645187

Offset: 0

Paul D. Hanna, Aug 26 2006

			Table of column 2 in matrix powers of triangle H=A121412 begins:
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, ...
Rearrangement of the upper part of the table forms A121432, which is
the number of subpartitions of partition [0,0,0,1,1,1,1,2,2,2,2,2,..]:
1,1,1, 1,2,3,4, 5,11,18,26,35, 45,101,169,250,345,455, 581,1305,...

Cf. A121429 (diagonal), A121431; rows: A121414, A121419, A121423; related tables: A121424, A121426; related triangles: A121412, A121416, A121420.

{T(n,k)=local(H=Mat(1), B); for(m=1, k+3, 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+3, 3])}

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].

cp's OEIS Frontend

A121428 Rectangular table, read by antidiagonals, where row n is equal to column 2 of matrix power A121412^(n+1) for n>=0.

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