A097356 -id:A097356 - cp's OEIS frontend

A173519 Number of partitions of n*(n+1)/2 into parts not greater than n.

1, 1, 2, 7, 23, 84, 331, 1367, 5812, 25331, 112804, 511045, 2348042, 10919414, 51313463, 243332340, 1163105227, 5598774334, 27119990519, 132107355553, 646793104859, 3181256110699, 15712610146876, 77903855239751, 387609232487489, 1934788962992123

Offset: 0

Reinhard Zumkeller, Feb 20 2010

nonn

a(n) is also the number of partitions of n^3 into n distinct parts <= n*(n+1). a(3) = 7: [4,11,12], [5,10,12], [6,9,12], [6,10,11], [7,8,12], [7,9,11], [8,9,10]. - Alois P. Heinz, Jan 25 2012

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..720 (terms 0..200 from Alois P. Heinz)

Cf. A066655, A097356, A258234.

Table[Length[IntegerPartitions[n(n + 1)/2, n]], {n, 10}] (* Alonso del Arte, Aug 12 2011 *)
Table[SeriesCoefficient[Product[1/(1-x^k),{k,1,n}],{x,0,n*(n+1)/2}],{n,0,20}] (* Vaclav Kotesovec, May 25 2015 *)

a(n)=
{
    local(tr=n*(n+1)/2, x='x+O('x^(tr+3)), gf);
    gf = 1 / prod(k=1,n, 1-x^k); /* g.f. for partitions into parts <=n */
    return( polcoeff( truncate(gf), tr ) );
} /* Joerg Arndt, Aug 14 2011 */

a(n) = A026820(A000217(n),n).

a(n) ~ c * d^n / n^2, where d = 5.4008719041181541524660911191042700520294... = A258234, c = 0.6326058791290010900659134913629203727... . - Vaclav Kotesovec, Sep 07 2014

More terms from D. S. McNeil, Aug 12 2011

A364526 Number of compositions (ordered partitions) of n into parts not greater than sqrt(n).

Original entry on oeis.org

1, 1, 1, 1, 5, 8, 13, 21, 34, 149, 274, 504, 927, 1705, 3136, 5768, 20569, 39648, 76424, 147312, 283953, 547337, 1055026, 2033628, 3919944, 11749641, 23099186, 45411804, 89277256, 175514464, 345052351, 678355061, 1333610936, 2621810068, 5154342880, 10133171296

Offset: 0

Ilya Gutkovskiy, Dec 22 2023

nonn

Index entries for sequences related to compositions

Cf. A011782, A097356.

a(n)=my(s=sqrtint(n), x='x+O('x^(n+1))); polcoef(Pol(Ser(1/(1-sum(j=1,s,x^j)))),n);
vector(33,n,a(n-1)) \\ Joerg Arndt, Dec 22 2023

a(n) = [x^n] 1 / (1 - Sum_{1 <= j <= sqrt(n)} x^j).

A097355 Number of partitions of n into parts not greater than floor(log_2(n)).

Original entry on oeis.org

1, 0, 1, 1, 3, 3, 4, 4, 10, 12, 14, 16, 19, 21, 24, 27, 64, 72, 84, 94, 108, 120, 136, 150, 169, 185, 206, 225, 249, 270, 297, 321, 831, 918, 1014, 1115, 1226, 1342, 1469, 1602, 1747, 1898, 2062, 2233, 2418, 2611, 2818, 3034, 3266, 3507, 3765, 4033, 4319, 4616, 4932

Offset: 0

Reinhard Zumkeller, Aug 08 2004

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000523, A000041, A097356.

a[n_] := Length[ IntegerPartitions[n, Floor[Log[2, n]]]]; Table[a[n], {n, 1, 54}] (* Jean-François Alcover, May 24 2012 *)

A194020 Number of partitions of n into parts not less than the integer part of the square root of n.

Original entry on oeis.org

1, 1, 2, 3, 2, 2, 4, 4, 7, 4, 5, 6, 9, 10, 13, 17, 11, 12, 16, 18, 24, 27, 34, 39, 50, 30, 36, 42, 50, 58, 70, 80, 95, 110, 129, 150, 96, 107, 126, 143, 167, 188, 221, 248, 288, 326, 376, 424, 491, 304, 346, 390, 443, 498, 565, 635, 719, 807, 911, 1022, 1153

Offset: 0

Reinhard Zumkeller, Aug 12 2011

			a(7) = #{7, 5+2, 4+3, 3+2+2} = 4;
a(8) = #{8, 6+2, 5+3, 4+4, 4+2+2, 3+3+2, 2+2+2+2} = 7;
a(9) = #{9, 6+3, 5+4, 3+3+3} = 4;
a(10) = #{10, 7+3, 6+4, 5+5, 4+3+3} = 5;
a(11) = #{11, 8+3, 7+4, 6+5, 5+3+3, 4+4+3} = 6.

Reinhard Zumkeller, Table of n, a(n) for n = 0..250

Cf. A097356, A000041, A000196.

a194020 n = p (a000196 n) n where
   p _  0 = 1
   p k m | m < k     = 0
         | otherwise = p k (m - k) + p (k+1) m

Table[Length[Select[IntegerPartitions[n], #[[-1]] >= Floor[Sqrt[n]] &]], {n, 60}] (* Alonso del Arte, Aug 12 2011 *)

A368502 Number of partitions of an n-set into blocks of size <= sqrt(n).

Original entry on oeis.org

1, 1, 1, 1, 10, 26, 76, 232, 764, 12644, 61136, 312676, 1680592, 9467680, 55704104, 341185496, 6631556521, 49294051497, 380306658250, 3039453750685, 25120541332271, 214363100120051, 1885987611214092, 17085579637664715, 159185637725413675, 3282701194678476257

Offset: 0

Ilya Gutkovskiy, Dec 27 2023

nonn

Cf. A000110, A097356, A368503.

Table[n! SeriesCoefficient[Exp[Sum[x^j/j!, {j, 1, Floor[Sqrt[n]]}]], {x, 0, n}], {n, 0, 25}]

a(n) = n! * [x^n] exp( Sum_{1 <= j <= sqrt(n)} x^j / j! ).

A316353 Number of partitions of positive integer n such that all parts are less than the square root of n.

Original entry on oeis.org

0, 1, 1, 1, 3, 4, 4, 5, 5, 14, 16, 19, 21, 24, 27, 30, 72, 84, 94, 108, 120, 136, 150, 169, 185, 427, 480, 540, 603, 674, 748, 831, 918, 1014, 1115, 1226, 2702, 3009, 3331, 3692, 4070, 4494, 4935, 5427, 5942, 6510, 7104, 7760, 8442, 18138, 19928, 21873, 23961, 26226, 28652

Offset: 1

Richard Locke Peterson, Jun 29 2018

nonn

This sequence itself is not a semigroup, but the set of all the partitions enumerated by this sequence does form a semigroup (actually a subsemigroup of the set of all partitions) with the following binary operation: let alpha = the partition (a,b,c,... [this is of course a finite list]) be the partition of the number N1 [that is, a + b + c + ... = N1] and let ALPHA = (A,B,C,...) be the partition of N2. Then the binary operation given by alpha*ALPHA = (a,b,c,...)*(A,B,C,...) = (aA,aB,aC,...,bA,bB,bC,...,cA,cB,cC,...) is a partition of the integer N1*N2. Furthermore, since any part x of alpha is less than the square root of N1, and likewise for any part Y of ALPHA, then the part xY is less than the square root of N1*N2, so the set is a subsemigroup of the semigroup of all partitions under the given operation. If the sole partition (1) of 1 is adjoined, the semigroup becomes a monoid.

			a(3)=1, since the partition (1,1,1) is the only partition of 3 with all parts less than the square root of 3 ~ 1.73.
a(6)=4, since there are only 4 allowable partitions: (1,1,1,1,1,1,1), (1,1,1,1,2), (1,1,2,2), and (2,2,2).

Alois P. Heinz, Table of n, a(n) for n = 1..20000 (first 123 terms from Robert G. Wilson v)

Cf. A000041 (the partition numbers), A097356 (with 'no greater' rather than less).

b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i<1, 0, b(n, i-1)+b(n-i, min(n-i, i))))
    end:
a:= n-> b(n, (r-> `if`(r*r>=n, r-1, r))(isqrt(n))):
seq(a(n), n=1..100);  # Alois P. Heinz, Aug 02 2018

Table[With[{s = Sqrt@ n}, Count[IntegerPartitions[n], ?(AllTrue[#, # < s &] &)]], {n, 53}] (* _Michael De Vlieger, Jul 22 2018 *)
f[n_] := Length@ IntegerPartitions[n, All, Range@ Sqrt[n - 1]]; Array[f, 50] (* Robert G. Wilson v, Jul 24 2018 *)
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1] + b[n - i, Min[n - i, i]]]];
a[n_] := b[n, Function[r, If[r*r >= n, r - 1, r]][Floor[Sqrt[n]]]];
Array[a, 100] (* Jean-François Alcover, May 30 2021, after Alois P. Heinz *)

a(n) = my(nb = 0); forpart(p=n, nb++, sqrtint(n)-issquare(n)); nb; \\ Michel Marcus, Jul 15 2018

log(a(n)) ~ log(A258268) * sqrt(n) - log(n). - Vaclav Kotesovec, May 30 2021

More terms from Michel Marcus, Jul 15 2018

A368868 Number of partitions of n into odd parts not greater than sqrt(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 8, 8, 8, 9, 29, 31, 33, 35, 37, 40, 42, 44, 47, 49, 52, 55, 57, 60, 63, 66, 69, 72, 75, 78, 82, 85, 88, 92, 292, 308, 324, 341, 358, 376, 395, 414, 434, 454, 475, 497, 519, 542, 566, 590, 615, 641, 667, 694, 722, 751

Offset: 0

Ilya Gutkovskiy, Jan 08 2024

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A000009, A097356.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+`if`(i::odd, b(n-i, min(n-i, i)), 0)))
    end:
a:= n-> b(n, floor(sqrt(n))):
seq(a(n), n=0..70);  # Alois P. Heinz, Jan 13 2024

Table[SeriesCoefficient[Product[1/(1 - Boole[OddQ[k]] x^k), {k, 1, Floor[Sqrt[n]]}], {x, 0, n}], {n, 0, 70}]

A369218 Number of partitions of n into prime power parts (including 1) not greater than sqrt(n).

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 4, 4, 5, 12, 14, 16, 19, 21, 24, 27, 64, 72, 84, 94, 108, 120, 136, 150, 169, 377, 427, 480, 540, 603, 674, 748, 831, 918, 1014, 1115, 1226, 1342, 1469, 1602, 1747, 1898, 2062, 2233, 2418, 2611, 2818, 3034, 3266, 7529, 8185, 8884, 9632, 10427, 11276, 12177

Offset: 0

Ilya Gutkovskiy, Jan 16 2024

nonn

Cf. A023893, A097356, A369219.

Table[SeriesCoefficient[Product[1/(1 - Boole[k == 1 || PrimePowerQ[k]] x^k), {k, 1, Floor[Sqrt[n]]}], {x, 0, n}], {n, 0, 55}]

A369219 Number of partitions of n into prime power parts (not including 1) not greater than sqrt(n).

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 1, 0, 1, 2, 2, 2, 3, 2, 3, 3, 10, 8, 12, 10, 14, 12, 16, 14, 19, 44, 50, 53, 60, 63, 71, 74, 83, 87, 96, 101, 111, 116, 127, 133, 145, 151, 164, 171, 185, 193, 207, 216, 232, 612, 656, 699, 748, 795, 849, 901, 960, 1017, 1081, 1144, 1214, 1282, 1358, 1433, 3620, 3845

Offset: 0

Ilya Gutkovskiy, Jan 16 2024

nonn

Cf. A023894, A097356, A369218.

Table[SeriesCoefficient[Product[1/(1 - Boole[PrimePowerQ[k]] x^k), {k, 1, Floor[Sqrt[n]]}], {x, 0, n}], {n, 0, 65}]

A368869 Number of partitions of n into prime parts not greater than sqrt(n).

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 1, 0, 1, 2, 2, 2, 3, 2, 3, 3, 3, 3, 4, 3, 4, 4, 4, 4, 5, 15, 16, 17, 18, 19, 21, 21, 23, 24, 25, 27, 28, 29, 31, 32, 34, 35, 37, 38, 40, 42, 43, 45, 47, 150, 158, 166, 175, 183, 193, 202, 212, 222, 232, 243, 254, 265, 277, 289, 301, 314, 327, 340, 354, 368, 383

Offset: 0

Ilya Gutkovskiy, Jan 08 2024

nonn

Cf. A000607, A097356.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+`if`(isprime(i), b(n-i, min(n-i, i)), 0)))
    end:
a:= n-> b(n, floor(sqrt(n))):
seq(a(n), n=0..70);  # Alois P. Heinz, Jan 13 2024

Table[SeriesCoefficient[Product[1/(1 - Boole[PrimeQ[k]] x^k), {k, 1, Floor[Sqrt[n]]}], {x, 0, n}], {n, 0, 70}]

cp's OEIS Frontend

A173519 Number of partitions of n*(n+1)/2 into parts not greater than n.

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A364526 Number of compositions (ordered partitions) of n into parts not greater than sqrt(n).

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A097355 Number of partitions of n into parts not greater than floor(log_2(n)).

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Mathematica

A194020 Number of partitions of n into parts not less than the integer part of the square root of n.

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Haskell

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A368502 Number of partitions of an n-set into blocks of size <= sqrt(n).

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Mathematica

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A316353 Number of partitions of positive integer n such that all parts are less than the square root of n.

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Maple

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A368868 Number of partitions of n into odd parts not greater than sqrt(n).

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Maple

Mathematica

A369218 Number of partitions of n into prime power parts (including 1) not greater than sqrt(n).

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Keywords

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Mathematica

A369219 Number of partitions of n into prime power parts (not including 1) not greater than sqrt(n).

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