A238016 -id:A238016 - cp's OEIS frontend

A094373 Expansion of (1-x-x^2)/((1-x)(1-2x)).

1, 2, 3, 5, 9, 17, 33, 65, 129, 257, 513, 1025, 2049, 4097, 8193, 16385, 32769, 65537, 131073, 262145, 524289, 1048577, 2097153, 4194305, 8388609, 16777217, 33554433, 67108865, 134217729, 268435457, 536870913, 1073741825, 2147483649, 4294967297, 8589934593

Offset: 0

Paul Barry, Apr 28 2004

Partial sum of 1,1,1,2,4,8,...
Binomial transform of abs(A073097).
Binomial transform is A094374.
Partial sums are in A006127. - Paul Barry, Aug 05 2004
An elephant sequence, see A175654. For the corner squares four A[5] vectors, with decimal values 2, 8, 32 and 128, lead to this sequence. For the central square these vectors lead to the companion sequence A011782. - Johannes W. Meijer, Aug 15 2010
This sequence has a(0) = 1 and for all n > 0, a(n) = 2^(n-1)+1. Consequently 2*a(n) >= a(n+1) for all n > 0 and the sequence is complete. - Frank M Jackson, Jan 29 2012
Row lengths of the triangle in A198069. - Reinhard Zumkeller, May 26 2013
Take A007843 and count the repeated values. The result is 1,1,2,1,3,1,2,1,4,1,2,1,3,1,2,1,5,.... Build a third sequence, where a(1) = 1 and a(n) equals the length (greater than 1) of the shortest palindromic subsequence of consecutive terms of the second sequence starting with a(n) of the second sequence. The third sequence starts 1,3,5,3,9,3,5,3,17,3,5,3,9,3,5,3,33,.... Conjecturally, in the third sequence: (1) the indices of the first occurrence of each value form the present sequence and (2) for n>1, a(n) is in the a(n-1)-th position. - Ivan N. Ianakiev, Aug 20 2019

			G.f. = 1 + 2*x + 3*x^2 + 5*x^3 + 9*x^4 + 17*x^5 + 33*x^6 + 65*x^7 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Eric Weisstein, Complete Sequence.
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Apart from the initial 1, identical to A000051.
Cf. A135225.
Column k=1 of A152977.
Row n=2 of A238016.

a:=[2,3];; for n in [3..40] do a[n]:=3*a[n-1]-2*a[n-2]; od; Concatenation([1], a); # G. C. Greubel, Nov 06 2019

[(2^n-0^n)/2+1: n in [0..40]]; // Vincenzo Librandi, Jun 10 2011

R:=PowerSeriesRing(Integers(), 35); Coefficients(R!( (1-x-x^2)/((1-x)*(1-2*x)))); // Marius A. Burtea, Oct 25 2019

1, seq((2^n - 0^n)/2 +1, n=1..40); # G. C. Greubel, Nov 06 2019

CoefficientList[Series[(1-x-x^2)/((1-x)*(1-2*x)), {x, 0, 40}], x] (* or *) Join[{1}, LinearRecurrence[{3, -2}, {2, 3}, 40]] (* Vladimir Joseph Stephan Orlovsky, Jan 22 2012 *)
a[ n_]:= If[n<0, 0, 1 + Quotient[2^n, 2]]; (* Michael Somos, May 26 2014 *)
a[ n_]:= SeriesCoefficient[(1-x-x^2)/((1-x)(1-2x)), {x, 0, n}]; (* Michael Somos, May 26 2014 *)
LinearRecurrence[{3,-2},{1,2,3},40] (* Harvey P. Dale, Aug 09 2015 *)

a(n)=2^n\2+1 \\ Charles R Greathouse IV, Apr 05 2013

Vec((1-x-x^2)/((1-x)*(1-2*x))+O(x^40)) \\ Charles R Greathouse IV, Apr 05 2013

[(2^n - 0^n)/2 + 1 for n in (0..40)] # G. C. Greubel, Nov 06 2019

a(n) = (2^n - 0^n)/2 + 1.
a(n) = 3*a(n-1) - 2*a(n-2).
a(2*n) = 2*a(2*n-1) - 1, n>0.
Row sums of triangle A135225. - Gary W. Adamson, Nov 23 2007
a(n) = A131577(n) + 1. - Paul Curtz, Aug 07 2008
a(n) = 2*a(n-1) - 1 for n>1, a(0)=1, a(1)=2. - Philippe Deléham, Sep 25 2009
E.g.f.: exp(x)*(1 + sinh(x)). - Arkadiusz Wesolowski, Aug 13 2012
G.f.: G(0), where G(k)= 1 + 2^k*x/(1 - x/(x + 2^k*x/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 26 2013
a(n) = 2^(n-1) +1 = A000051(n-1) for n>0. - M. F. Hasler, Sep 22 2013

A238010 Number A(n,k) of partitions of k^n into parts that are at most n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 3, 1, 1, 0, 1, 5, 10, 1, 1, 0, 1, 9, 75, 64, 1, 1, 0, 1, 13, 374, 4410, 831, 1, 1, 0, 1, 19, 1365, 123464, 1366617, 26207, 1, 1, 0, 1, 25, 3997, 1736385, 393073019, 2559274110, 2239706, 1, 1

Offset: 0

Alois P. Heinz, Feb 16 2014

In general, column k>=2 is asymptotic to k^(n*(n-1)) / (n!*(n-1)!). - Vaclav Kotesovec, Jun 05 2015

			A(3,2) = 10: 332, 2222, 3221, 3311, 22211, 32111, 221111, 311111, 2111111, 11111111.
A(2,3) = 5: 22221, 222111, 2211111, 21111111, 111111111.
A(2,4) = 9: 22222222, 222222211, 2222221111, 22222111111, 222211111111, 2221111111111, 22111111111111, 211111111111111, 1111111111111111.
Square array A(n,k) begins:
  0, 0,   0,       0,         0,           0, ...
  1, 1,   1,       1,         1,           1, ...
  1, 1,   3,       5,         9,          13, ...
  1, 1,  10,      75,       374,        1365, ...
  1, 1,  64,    4410,    123464,     1736385, ...
  1, 1, 831, 1366617, 393073019, 33432635477, ...

Alois P. Heinz, Antidiagonals n = 0..43, flattened
A. V. Sills and D. Zeilberger, Formulae for the number of partitions of n into at most m parts (using the quasi-polynomial ansatz) (arXiv:1108.4391 [math.CO])

Columns k=0+1,2-10 give: A057427, A237998, A238560, A238561, A238562, A238563, A238564, A238565, A238566, A238567.
Rows n=0-2 give: A000004, A000012, A080827.
Main diagonal gives A238000.
Cf. A238012, A238016.

A[n_, k_] := SeriesCoefficient[Product[1/(1-x^j), {j, 1, n}], {x, 0, k^n}]; A[0, 0] = 0; Table[A[n-k, k], {n, 0, 9}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Feb 17 2017 *)

A(n,k) = [x^(k^n)] Product_{j=1..n} 1/(1-x^j).

A206226 Number of partitions of n^2 into parts not greater than n.

Original entry on oeis.org

1, 1, 3, 12, 64, 377, 2432, 16475, 116263, 845105, 6292069, 47759392, 368379006, 2879998966, 22777018771, 181938716422, 1465972415692, 11902724768574, 97299665768397, 800212617435074, 6617003142869419, 54985826573015541, 458962108485797208, 3846526994743330075

Offset: 0

Paul D. Hanna, Feb 05 2012

nonn

Also the number of partitions of n^2 using n or fewer numbers. Thus for n=3 one has: 9; 1,8; 2,7; 3,6; 4,5; 1,1,7; 1,2,6; 1,3,5; 1,4,4; 2,2,5; 2,3,4; 3,3,3. - J. M. Bergot, Mar 26 2014 [computations done by Charles R Greathouse IV]
The partitions in the comments above are the conjugates of the partitions in the definition. By conjugation we have: "partitions into parts <= m" are equinumerous with "partitions into at most m parts". - Joerg Arndt, Mar 31 2014
From Vaclav Kotesovec, May 25 2015: (Start)
In general, "number of partitions of j*n^2 into parts that are at most n" is (for j>0) asymptotic to c(j) * d(j)^n / n^2, where c(j) and d(j) are a constants.
-------
j  c(j)
1  0.1582087202672504149766310999238...
2  0.0794245035465730707705885572860...
3  0.0530017980244665552354063060738...
4  0.0397666338404544208556554596295...
5  0.0318193213988281353709268311928...
...
17 0.0093617308583114626385718275875...
   c(j) for big j asymptotically approaches 1 / (2*Pi*j).
---------
j    d(j)
1    9.15337019245412246194853029240... = A258268
2   16.57962120993269533568313969522...
3   23.98280768122086592445663786762...
4   31.37931997386325137074644287711...
5   38.77298550971449870728474612568...
...
17 127.45526806942537991146993713837...
   d(j) for big j asymptotically approaches j * exp(2).
(End)
d(j) = r^(2*j+1)/(r-1), where r is the root of the equation polylog(2, 1-r) + (j+1/2)*log(r)^2 = 0. - Vaclav Kotesovec, Jun 11 2015

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..382 (first 150 terms from Alois P. Heinz)

Cf. A173519, A206227, A206240, A107379, A258268.
Column k=2 of A238016.
Cf. A258296 (j=2), A258293 (j=3), A258294 (j=4), A258295 (j=5).

T:= proc(n, k) option remember;
      `if`(n=0 or k=1, 1, T(n, k-1) + `if`(k>n, 0, T(n-k, k)))
    end:
seq(T(n^2, n), n=0..20); # Vaclav Kotesovec, May 25 2015 after Alois P. Heinz

Table[SeriesCoefficient[Product[1/(1-x^k),{k,1,n}],{x,0,n^2}],{n,0,20}] (* Vaclav Kotesovec, May 25 2015 *)
(* A program to compute the constants d(j) *) Table[r^(2*j+1)/(r-1) /.FindRoot[-PolyLog[2,1-r] == (j+1/2)*Log[r]^2, {r, E}, WorkingPrecision->60], {j, 1, 5}] (* Vaclav Kotesovec, Jun 11 2015 *)

{a(n)=polcoeff(prod(k=1,n,1/(1-x^k+x*O(x^(n^2)))),n^2)}
for(n=0,25,print1(a(n),", "))

a(n) = [x^(n^2)] Product_{k=1..n} 1/(1 - x^k).

a(n) ~ c * d^n / n^2, where d = 9.1533701924541224619485302924013545... = A258268, c = 0.1582087202672504149766310999238742... . - Vaclav Kotesovec, Sep 07 2014

A238608 Number of partitions of n^3 into parts that are at most n.

Original entry on oeis.org

1, 1, 5, 75, 2280, 106852, 6889527, 569704489, 57733506640, 6944433285769, 968356321790171, 153738253618009045, 27396489338187214000, 5417302365503826145732, 1177436831956414016252071, 279074576444362385794783853, 71649589941044468875380333533

Offset: 0

Alois P. Heinz, Mar 01 2014

nonn

In general, "number of partitions of j*n^3 into parts that are at most n" is (for j>0) asymptotic to exp(2*n + 1/(4*j)) * n^(n-3) * j^(n-1) / (2*Pi). - Vaclav Kotesovec, May 25 2015

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..122 (terms 0..70 from Alois P. Heinz)

Column k=3 of A238016.

Cf. A258302 (j=2), A258303 (j=3), A258304 (j=4), A258305 (j=5).

T:=proc(n,k) option remember; `if`(n=0 or k=1, 1, T(n,k-1) + `if`(nVaclav Kotesovec, May 25 2015 after Alois P. Heinz

a[n_] := SeriesCoefficient[1/QPochhammer[q, q, n], {q, 0, n^3}]; Table[ a[n], {n, 0, 20}] (* Jean-François Alcover, Dec 03 2015 *)

a(n) = [x^(n^3)] Product_{j=1..n} 1/(1-x^j).

a(n) ~ exp(2*n + 1/4) * n^(n-3) / (2*Pi). - Vaclav Kotesovec, May 25 2015

A238000 Number of partitions of n^n into parts that are at most n.

Original entry on oeis.org

0, 1, 3, 75, 123464, 33432635477, 2561606354507677872, 85980297709044488588773397089, 1841159754991692001851990839259642586671980, 34687845413783594101366282545316028561007822069601179170488

Offset: 0

Alois P. Heinz, Feb 16 2014

nonn

			a(1) = 1: 1.
a(2) = 3: 22, 211, 1111.
a(3) = 75: 333333333, ..., 111111111111111111111111111.

Alois P. Heinz, Table of n, a(n) for n = 0..27
A. V. Sills and D. Zeilberger, Formulae for the number of partitions of n into at most m parts (using the quasi-polynomial ansatz) (arXiv:1108.4391 [math.CO])

Main diagonal of A238010 and A238016.

Cf. A236810, A237512, A237998, A237999, A238001.

a[n_] := SeriesCoefficient[Product[1/(1 - x^j), {j, 1, n}], {x, 0, n^n}];
a[0] = 0;
Table[a[n], {n, 0, 5}] (* Jean-François Alcover, Nov 03 2018 *)

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

a(n) ~ exp(2*n) * n^(n*(n-3)) / (2*Pi). - Vaclav Kotesovec, May 25 2015

A258268 Decimal expansion of a constant related to A206226.

Original entry on oeis.org

9, 1, 5, 3, 3, 7, 0, 1, 9, 2, 4, 5, 4, 1, 2, 2, 4, 6, 1, 9, 4, 8, 5, 3, 0, 2, 9, 2, 4, 0, 1, 3, 5, 4, 5, 4, 0, 0, 7, 3, 3, 2, 7, 2, 0, 4, 1, 2, 1, 8, 4, 8, 8, 4, 9, 6, 8, 9, 2, 6, 3, 2, 0, 1, 4, 7, 6, 1, 3, 8, 3, 7, 6, 6, 8, 9, 5, 7, 3, 1, 6, 2, 3, 9, 1, 5, 1, 9, 0, 2, 5, 5, 8, 7, 9, 5, 1, 9, 2, 8, 4, 5, 3, 8, 9

Offset: 1

Vaclav Kotesovec, May 25 2015

			9.153370192454122461948530292401354540073...

Cf. A206226, A206227, A206240, A258234, A107379, A173519, A238016, A258789.

r^3/(r-1) /.FindRoot[-PolyLog[2, 1-r] == 3*Log[r]^2/2, {r, E}, WorkingPrecision->120] (* Vaclav Kotesovec, Jun 11 2015 *)

Equals limit n->infinity A206226(n)^(1/n).
Equals limit n->infinity A206227(n)^(1/n).
Equals limit n->infinity A206240(n)^(1/n).

More digits from Vaclav Kotesovec, Jun 10 2015

A258670 Number of partitions of (2*n)! into parts that are at most n.

Original entry on oeis.org

0, 1, 13, 43561, 455366036161, 60209252317216962943201, 291857679749953126623181556402787323521, 120972618144269517756284629487432992029777542693069847287041

Offset: 0

Vaclav Kotesovec, Jun 07 2015

nonn

Conjecture: If f(n) >= O(n^4) then "number of partitions of f(n) into parts that are at most n" is asymptotic to f(n)^(n-1) / (n!*(n-1)!). For the examples see A238016 and A238010.

Vaclav Kotesovec, Table of n, a(n) for n = 0..21
G. J. Rieger, Über Partitionen, Mathematische Annalen (1959), Volume: 138, page 356-362
A. V. Sills and D. Zeilberger, Formulae for the number of partitions of n into at most m parts (using the quasi-polynomial ansatz) (arXiv:1108.4391 [math.CO])

Cf. A236810, A237998, A238000, A238010, A238016, A258668, A258669, A258671.

a(n) ~ (2*n)!^(n-1) / (n!*(n-1)!).

A347615 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) is the number of partitions of n^k.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 5, 3, 1, 1, 1, 22, 30, 5, 1, 1, 1, 231, 3010, 231, 7, 1, 1, 1, 8349, 18004327, 1741630, 1958, 11, 1, 1, 1, 1741630, 133978259344888, 365749566870782, 3163127352, 17977, 15, 1, 1, 1, 4351078600, 233202632378520643600875145, 61847822068260244309086870983975, 1606903190858354689128371, 15285151248481, 173525, 22, 1

Offset: 0

Seiichi Manyama, Sep 08 2021

			Square array begins:
  1, 1,   1,       1,               1, ...
  1, 1,   1,       1,               1, ...
  1, 2,   5,      22,             231, ...
  1, 3,  30,    3010,        18004327, ...
  1, 5, 231, 1741630, 365749566870782, ...

Columns k=0..3 give A000012, A000041, A072213, A128854.
Rows n=0+1, 2-10 give A000012, A068413, A248728, A068413(2*n), A248730, A248732, A248734, A068413(3*n), A248728(2*n), A070177.
Main diagonal gives A347607.
Cf. A238016, A347617, A347618, A347621.

PARI
```
T(n, k) = numbpart(n^k);
```

T(n,k) = A000041(n^k).

A258672 Number of partitions of n*2^n into parts that are at most n.

Original entry on oeis.org

0, 1, 5, 61, 2280, 273052, 110537709, 156456474138, 790541795804221, 14445283925963101577, 963056085414756870071490, 235864774408401842540220265704, 213426797830699546133563821747980513, 717147073290996884137625501875655000693923

Offset: 0

Vaclav Kotesovec, Jun 07 2015

nonn

Conjecture: If f(n) >= O(n^4) then "number of partitions of f(n) into parts that are at most n" is asymptotic to f(n)^(n-1) / (n!*(n-1)!). For the examples see A238016 and A238010.

Vaclav Kotesovec, Table of n, a(n) for n = 0..59
A. V. Sills and D. Zeilberger, Formulae for the number of partitions of n into at most m parts (using the quasi-polynomial ansatz) (arXiv:1108.4391 [math.CO])

Cf. A236810, A237998, A238000, A238010, A238016.

a(n) ~ n^n * 2^(n*(n-1)) / (n!)^2.

A238609 Number of partitions of n^4 into parts that are at most n.

Original entry on oeis.org

1, 1, 9, 588, 123464, 55567352, 44056912182, 54667189410224, 98149884074667116, 241192889005578902877, 778400276435728381405745, 3195674736701993615997749350, 16272552341081798500863569890566, 100683204917037438858515986247835992

Offset: 0

Alois P. Heinz, Mar 01 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..70

Column k=4 of A238016.

a(n) = [x^(n^4)] Product_{j=1..n} 1/(1-x^j).

a(n) ~ exp(2*n) * n^(2*n-4) / (2*Pi). - Vaclav Kotesovec, May 25 2015

cp's OEIS Frontend

A094373 Expansion of (1-x-x^2)/((1-x)*(1-2*x)).

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A238010 Number A(n,k) of partitions of k^n into parts that are at most n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A206226 Number of partitions of n^2 into parts not greater than n.

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A238608 Number of partitions of n^3 into parts that are at most n.

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A238000 Number of partitions of n^n into parts that are at most n.

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A258268 Decimal expansion of a constant related to A206226.

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A258670 Number of partitions of (2*n)! into parts that are at most n.

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A094373 Expansion of (1-x-x^2)/((1-x)(1-2x)).