A115728 -id:A115728 - cp's OEIS frontend

A109055 To compute a(n) we first write down 3^n 1's in a row. Each row takes the rightmost 3rd part of the previous row and each element in it equals sum of the elements of the previous row starting with the first of the rightmost 3rd part. The single element in the last row is a(n).

1, 1, 3, 24, 541, 35649, 6979689, 4085743032, 7166723910237, 37698139930450365, 594816080266215640710, 28154472624850002001979592, 3997853576535778666975681355079, 1703042427700923785323670557504832751, 2176429411666209822350337722381643148477248

Offset: 0

Augustine O. Munagi, Jun 17 2005

nonn

Comment from Franklin T. Adams-Watters, Jul 13 2006: This is the number of subpartitions of the sequence 3^n-1. As such it can also be computed adding forward, with 3^n terms in the n-th line:
..........................................................................
1 1.......................................................................
2.3.3..3..3..3..3..3......................................................
3.6.9.12.15.18.21.24.24.24.24.24.24.24.24.24.24.24.24.24.24.24.24.24.24.24

			For example, for n=3 the array looks like this:
1..1..1..1..1........1..1..1..1..1..1..1..1..1..1
........................1..2..3..4..5..6..7..8..9
..........................................7.15.24
...............................................24
Therefore a(3)=24.

Alois P. Heinz, Table of n, a(n) for n = 0..65

Cf. A107354, A109056, A109057, A109058, A109059, A109060, A109061, A109062.
Cf. A115728, A115729.
Column k=3 of A355576.

proc(n::nonnegint) local f,a; if n=0 or n=1 then return 1; end if; f:=L->[seq(add(L[i],i=2*nops(L)/3+1..j),j=2*nops(L)/3+1..nops(L))]; a:=f([seq(1,j=1..3^n)]); while nops(a)>3 do a:=f(a) end do; a[3]; end proc;

A[n_, k_] := A[n, k] = If[n == 0, 1, -Sum[A[j, k]*(-1)^(n - j)*Binomial[If[j == 0, 1, k^j], n - j], {j, 0, n - 1}]];
a[n_] := A[n, 3];
Table[a[n], {n, 0, 14}] (* Jean-François Alcover, Apr 01 2024, after Alois P. Heinz in A355576 *)

More terms from Paul D. Hanna

A177450 G.f.: Sum_{n>=0} a(n)*x^n/(1+x)^(n^2+n) = 1+x.

Original entry on oeis.org

1, 1, 2, 9, 70, 805, 12480, 245847, 5909338, 168310515, 5556486450, 209003251240, 8835266400450, 415094928861530, 21473740362658640, 1213683089969940075, 74446121738526773490, 4927385997649620215895, 350145746700442604768346

Offset: 0

Paul D. Hanna, May 09 2010

nonn

			1+x = 1 + 1*x/(1+x)^2 + 2*x^2/(1+x)^6 + 9*x^3/(1+x)^12 + 70*x^4/(1+x)^20 + 805*x^5/(1+x)^30 +...
1/(1-x) = 1 + 1*x*(1-x) + 2*x^2*(1-x)^4 + 9*x^3*(1-x)^9 + 70*x^4*(1-x)^16 + 805*x^5*(1-x)^25 +...
Also forms the final terms in rows of the triangle where row n+1 equals the partial sums of row n with the final term repeated 2(n+1) times, starting with a '1' in row 0, as illustrated by:
  1;
  1, 1;
  1, 2,  2,  2,  2;
  1, 3,  5,  7,  9,  9,   9,   9,   9,   9;
  1, 4,  9, 16, 25, 34,  43,  52,  61,  70,  70,  70,  70,  70,  70,  70,  70;
  1, 5, 14, 30, 55, 89, 132, 184, 245, 315, 385, 455, 525, 595, 665, 735, 805, 805, 805, 805, 805, 805, 805, 805, 805, 805;
  ...

Alois P. Heinz, Table of n, a(n) for n = 0..356

Cf. A107877, A177447, A177448, A177449, A209440.

a:= proc(n) option remember; `if`(n=0, 1, -add(a(j)
      *(-1)^(n-j)*binomial(1+ j^2, n-j), j=0..n-1))
    end:
seq(a(n), n=0..19);  # Alois P. Heinz, Jul 08 2022

{a(n)=local(F=1/(1+x+x*O(x^n)));polcoeff(1+x-sum(k=0,n-1,a(k)*x^k*F^(k*(k+1))),n)}

G.f.: Sum_{n>=0} a(n)*x^n*(1-x)^(n^2) = 1/(1-x).
G.f.: Sum_{n>=0} a(n)*x^n*C(-x)^(n^2+2n) = 1/C(-x) where C(x) is the Catalan function of A000108.
a(n) = number of subpartitions of partition consisting of the first n square numbers starting with zero for n>0; e.g., a(4) = subp([0,1,4,9]) = 70. See A115728 for the definition of subpartitions.

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).

A115723 Table of partitions of n with maximum rectangle k.

Original entry on oeis.org

1, 0, 2, 0, 1, 2, 0, 0, 2, 3, 0, 0, 1, 4, 2, 0, 0, 0, 5, 2, 4, 0, 0, 0, 3, 4, 6, 2, 0, 0, 0, 1, 4, 11, 2, 4, 0, 0, 0, 0, 3, 14, 4, 6, 3, 0, 0, 0, 0, 1, 15, 6, 12, 4, 4, 0, 0, 0, 0, 0, 13, 8, 18, 9, 6, 2, 0, 0, 0, 0, 0, 8, 10, 25, 14, 12, 2, 6, 0, 0, 0, 0, 0, 4, 9, 30, 22, 20, 4, 10, 2

Offset: 1

Franklin T. Adams-Watters, Mar 11 2006

T(n,k)=0 if n > A006218(k).

			The table starts:
  1;
  0, 2;
  0, 1, 2;
  0, 0, 2, 3;
  0, 0, 1, 4, 2;
  0, 0, 0, 5, 2,  4;
  0, 0, 0, 3, 4,  6, 2;
  0, 0, 0, 1, 4, 11, 2,  4;
  0, 0, 0, 0, 3, 14, 4,  6, 3;
  0, 0, 0, 0, 1, 15, 6, 12, 4, 4;
  ...

Alois P. Heinz, Rows n = 1..141, flattened
Eric Weisstein's World of Mathematics, Ferrers Diagram.

Cf. A000005 (diagonal), A000041 (row sums), A061017 (column indices of leftmost nonzero elements), A115724 (column sums), A115727, A115728, A006218, A182099.

Sum_{k=1..n} k * T(n,k) = A182099(n).

A115725 Number of partitions with maximum rectangle <= n.

Original entry on oeis.org

1, 2, 5, 10, 26, 42, 118, 171, 389, 692, 1442, 1854, 5534, 6895, 11910, 21116, 44278, 52568, 118734, 138670, 300326, 492507, 728514, 829244, 2167430, 2987124, 4167602, 6092588, 11308432, 12554900, 29925267, 33023589, 57950313, 81424281, 106214784, 148101088

Offset: 0

Franklin T. Adams-Watters, Mar 11 2006

nonn

A partition has maximum rectangle <= n iff it is a subpartition of row n of A010766.

			The 10 partitions with maximum rectangle <= 3: 0: []; 1: [1]; 2: [2], [1^2], [2,1]; 3: [3], [1^3], [3,1], [2,1^2], [3,1^2].

Alois P. Heinz, Table of n, a(n) for n = 0..100
Eric Weisstein's World of Mathematics, Ferrers Diagram.

Cf. A115728, A115729, A115724, A010766.

a(n) = subpart([A115728 (or A115729), [] is row n of A010766.

a(n) = Sum_{k>=0} A182114(k,n). - Alois P. Heinz, Nov 02 2012

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].

A177448 G.f.: Sum_{n>=0} a(n)x^n/(1+x)^(2n^2) = 1+x.

Original entry on oeis.org

1, 1, 2, 13, 166, 3324, 92718, 3354712, 150206430, 8050991676, 504049958320, 36172232930282, 2931474921768206, 265078092222575572, 26480336590135734816, 2898139377307388441520, 345055687960080723910286

Offset: 0

Paul D. Hanna, May 09 2010

nonn

			1+x = 1 + 1*x/(1+x)^2 + 2*x^2/(1+x)^8 + 13*x^3/(1+x)^18 + 166*x^4/(1+x)^32 + 3324*x^5/(1+x)^50 + 92718*x^6/(1+x)^72 +...

Cf. A177447, A177449, A177450.

{a(n)=local(F=1/(1+x+x*O(x^n)));polcoeff(1+x-sum(k=0,n-1,a(k)*x^k*F^(2*k^2)),n)}

a(n) = number of subpartitions of the partition [0,1,6,15,28,...,2(n-1)^2-(n-1)] for n>0 with a(0)=1. See A115728 for the definition of subpartitions.

A177449 G.f.: Sum_{n>=0} a(n)x^n/(1+x)^(3n^2) = 1+x.

Original entry on oeis.org

1, 1, 3, 30, 586, 17865, 756285, 41440056, 2805638310, 227131872654, 21459076173105, 2322336372705030, 283667666439112350, 38643426990067599005, 5813534115429573742587, 957883907138024944675200

Offset: 0

Paul D. Hanna, May 09 2010

nonn

			1+x = 1 + 1*x/(1+x)^3 + 3*x^2/(1+x)^12 + 30*x^3/(1+x)^27 + 586*x^4/(1+x)^48 + 17865*x^5/(1+x)^75 + 756285*x^6/(1+x)^108 +...

Cf. A177447, A177448, A177450.

{a(n)=local(F=1/(1+x+x*O(x^n)));polcoeff(1+x-sum(k=0,n-1,a(k)*x^k*F^(3*k^2)),n)}

a(n) = number of subpartitions of the partition [0,2,10,24,44,...,3(n-1)^2-(n-1)] for n>0 with a(0)=1. See A115728 for the definition of subpartitions.

cp's OEIS Frontend

A109055 To compute a(n) we first write down 3^n 1's in a row. Each row takes the rightmost 3rd part of the previous row and each element in it equals sum of the elements of the previous row starting with the first of the rightmost 3rd part. The single element in the last row is a(n).

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A177450 G.f.: Sum_{n>=0} a(n)*x^n/(1+x)^(n^2+n) = 1+x.

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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).

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A115723 Table of partitions of n with maximum rectangle k.

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A115725 Number of partitions with maximum rectangle <= n.

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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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PARI

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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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A177448 G.f.: Sum_{n>=0} a(n)x^n/(1+x)^(2n^2) = 1+x.

A177449 G.f.: Sum_{n>=0} a(n)x^n/(1+x)^(3n^2) = 1+x.