A330643 -id:A330643 - cp's OEIS frontend

A330642 a(n) is the number of partitions of n with Durfee square of size <= 4.

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 490, 627, 792, 1002, 1255, 1575, 1957, 2434, 3005, 3708, 4545, 5568, 6779, 8245, 9974, 12046, 14478, 17372, 20747, 24732, 29360, 34782, 41045, 48337, 56716, 66410, 77498, 90247, 104763, 121366, 140181, 161590, 185755

Offset: 0

Omar E. Pol, Dec 24 2019

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-1,-4,4,4,2,0,-10,0,2,4,4,-4,-1,-2,1,2,-1).

Cf. A000041, A006918, A008805, A028310, A115994, A115720, A117485, A117486, A330640, A330641, A330643.

seq(n) = Vec(sum(k=0, 4, x^(k^2)/prod(j=1, k, 1 - x^j)^2) + O(x*x^n)) \\ Andrew Howroyd, Dec 27 2024

a(n) = A000041(n), 0 <= n <= 24.
a(n) = A330641(n), 0 <= n <= 15.
a(n) = A330641(n) + A117486(n-16), n >= 16.
a(n) = n + A006918(n-3) + A117485(n) + A117486(n-16), n >= 16.
From Colin Barker, Jan 01 2020: (Start)
G.f.: (1 - x - x^2 + 3*x^5 - x^7 - 2*x^8 - 2*x^9 + 3*x^10 + x^11 + x^12 - x^13 - 2*x^14 + x^15 + x^17 - x^19 + x^20) / ((1 - x)^8*(1 + x)^4*(1 + x^2)^2*(1 + x + x^2)^2).
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) - a(n-4) - 4*a(n-5) + 4*a(n-6) + 4*a(n-7) + 2*a(n-8) - 10*a(n-10) + 2*a(n-12) + 4*a(n-13) + 4*a(n-14) - 4*a(n-15) - a(n-16) - 2*a(n-17) + a(n-18) + 2*a(n-19) - a(n-20) for n>20.
(End)
G.f.: Sum_{k=0..4} x^(k^2)/(Product_{j=1..k} (1 - x^j))^2. - Andrew Howroyd, Dec 27 2024

A330376 Irregular triangle read by rows: T(n,k) is the total number of parts in all partitions of n with Durfee square of size k (n>=1; 1<=k<=floor(sqrt(n))).

Original entry on oeis.org

1, 3, 6, 10, 2, 15, 5, 21, 14, 28, 26, 36, 50, 45, 80, 3, 55, 130, 7, 66, 190, 19, 78, 280, 41, 91, 385, 80, 105, 532, 143, 120, 700, 248, 136, 924, 399, 4, 153, 1176, 627, 9, 171, 1500, 949, 24, 190, 1860, 1397, 51, 210, 2310, 2003, 107, 231, 2805, 2823, 193

Offset: 1

Omar E. Pol, Dec 22 2019

			Triangle begins:
   1;
   3;
   6;
  10,  2;
  15,  5;
  21, 14;
  28, 26;
  36, 50;
  45, 80, 3;

Andrew Howroyd, Table of n, a(n) for n = 1..1799 (rows 1..200)
Eric Weisstein's World of Mathematics, Durfee Square

Row sums give A006128, n >= 1.
Column 1 gives A000217, n >= 1.
Cf. A114089, A115721, A115722, A115994, A116858, A118198, A208474, A330369, A330640, A330641, A330642, A330643.
Cf. A330369.

\\ by enumeration over partitions.
ds(p)={for(i=2, #p, if(p[#p+1-i]Andrew Howroyd, Feb 02 2022

\\ by generating function.
P(n,k,y)={1/prod(j=1, k, 1 - y*x^j + O(x*x^n))}
T(n,k)={my(r=n-k^2); if(r<0, 0, subst(deriv(polcoef(y^k*P(r,k,1)*P(r,k,y), r)), y, 1))}
{ for(n=1, 10, print(vector(sqrtint(n), k, T(n,k)))) } \\ Andrew Howroyd, Feb 02 2022

Terms a(10) and beyond from Andrew Howroyd, Feb 02 2022

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A330642 a(n) is the number of partitions of n with Durfee square of size <= 4.

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A330376 Irregular triangle read by rows: T(n,k) is the total number of parts in all partitions of n with Durfee square of size k (n>=1; 1<=k<=floor(sqrt(n))).

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