A303070 -id:A303070 - cp's OEIS frontend

A270915 Decimal expansion of a constant related to the asymptotics of A008485.

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

Offset: 1

Vaclav Kotesovec, Mar 25 2016

			5.352701333486642687772415814165327879851483271286947097319690756...

Vaclav Kotesovec, Table of n, a(n) for n = 1..500

Cf. A008485, A109084, A109085, A206228, A270914, A303070, A304625, A327279.

RealDigits[1/r /. FindRoot[{s == 1/QPochhammer[r*s], QPochhammer[r*s] + r*s*Derivative[0, 1][QPochhammer][r*s, r*s] == (Log[1 - r*s] + QPolyGamma[0, 1, r*s]) / (s*Log[r*s])}, {r, 1/5}, {s, 1}, WorkingPrecision -> 120], 10, 105][[1]] (* Vaclav Kotesovec, Sep 26 2023 *)

Equals limit n->infinity A008485(n)^(1/n).

A210764 Square array T(n,k), n>=0, k>=0, read by antidiagonals in which column k gives the partial sums of column k of A144064.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 7, 8, 4, 1, 1, 12, 18, 13, 5, 1, 1, 19, 38, 35, 19, 6, 1, 1, 30, 74, 86, 59, 26, 7, 1, 1, 45, 139, 194, 164, 91, 34, 8, 1, 1, 67, 249, 415, 416, 281, 132, 43, 9, 1, 1, 97, 434, 844, 990, 787, 447, 183, 53, 10, 1

Offset: 0

Omar E. Pol, Jun 27 2012

It appears that row 2 is A034856.
Observation:
Column 1 is the EULER transform of 2,1,1,1,1,1,1,1...
Column 2 is the EULER transform of 3,2,2,2,2,2,2,2...

			Array begins:
1,   1,   1,   1,   1,   1,   1,   1,   1,   1,   1,
1,   2,   3,   4,   5,   6,   7,   8,   9,  10,
1,   4,   8,  13,  19,  26,  34,  43,  53,
1,   7,  18,  35,  59,  91, 132, 183,
1,  12,  38,  86, 164, 281, 447,
1,  19,  74, 194, 416, 787,
1,  30, 139, 415, 990,
1,  45, 249, 844,
1,  67, 434,
1,  97,
1,

Alois P. Heinz, Rows n = 0..140, flattened

Columns (0-3): A000012, A000070, A000713, A210843.
Rows (0-1): A000012, A000027.
Main diagonal gives A303070.
Cf. A000007, A000041, A005758, A006922, A000712, A000716, A023003-A023021, A144064, A195825, A211970.

with(numtheory):
etr:= proc(p) local b;
        b:= proc(n) option remember; `if`(n=0, 1,
              add(add(d*p(d), d=divisors(j))*b(n-j), j=1..n)/n)
            end
      end:
A:= (n, k)-> etr(j-> k +`if`(j=1, 1, 0))(n):
seq(seq(A(d-k, k), k=0..d), d=0..14); # Alois P. Heinz, May 20 2013

etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[ j]}]*b[n-j], {j, 1, n}]/n]; b]; A[n_, k_] := etr[Function[{j}, k + If[j == 1, 1, 0]]][n]; Table[Table[A[d-k, k], {k, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Mar 05 2015, after Alois P. Heinz *)

A303071 a(n) = [x^n] (1/(1 - x))*Product_{k>=1} (1 + x^k)^n.

Original entry on oeis.org

1, 2, 6, 23, 90, 362, 1491, 6225, 26242, 111479, 476466, 2046464, 8825559, 38191467, 165751529, 721177328, 3144703234, 13739010855, 60127642329, 263545670385, 1156732481150, 5083320593976, 22364017244278, 98491038664903, 434160710647831, 1915482295831037, 8457663096970431

Offset: 0

Ilya Gutkovskiy, Apr 18 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..300
Index entries for sequences related to partitions

Cf. A036469, A270913, A286335, A303070.

Table[SeriesCoefficient[1/(1 - x) Product[(1 + x^k)^n, {k, 1, n}], {x, 0, n}], {n, 0, 26}]
Table[SeriesCoefficient[1/(1 - x) Exp[n Sum[(-1)^(k + 1) x^k/(k (1 - x^k)), {k, 1, n}]], {x, 0, n}], {n, 0, 26}]

a(n) = [x^n] (1/(1 - x))*exp(n*Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k))).
a(n) = Sum_{j=0..n} A286335(j,n).
a(n) ~ c * d^n / sqrt(n), where d = A270914 = 4.5024767476173544877385939327007... and c = 0.44252758868364961050787771300805... - Vaclav Kotesovec, May 19 2018

A303914 a(n) = [x^n] (1/(1 - x))Product_{k>=1} 1/(1 - nx^k).

Original entry on oeis.org

1, 2, 9, 55, 465, 5051, 69265, 1147287, 22307905, 497211049, 12484203601, 348391613615, 10691846920081, 357749800027465, 12958472141161457, 505088781523073326, 21076091000708067585, 937322034938743608556, 44256147057318887809993, 2210813717869831566759857, 116492226446226314836976401

Offset: 0

Ilya Gutkovskiy, May 02 2018

nonn

Index entries for sequences related to partitions

Cf. A000070, A124577, A246935, A303070.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1, k) +`if`(i>n, 0, k*b(n-i, i, k))))
    end:
a:= n-> add(b(j$2, n), j=0..n):
seq(a(n), n=0..20);  # Alois P. Heinz, May 02 2018

Table[SeriesCoefficient[1/(1 - x) Product[1/(1 - n x^k), {k, 1, n}], {x, 0, n}], {n, 0, 20}]
Table[SeriesCoefficient[1/(1 - x) Exp[Sum[n^k x^k/(k (1 - x^k)), {k, 1, n}]], {x, 0, n}], {n, 0, 20}]

a(n) = [x^n] (1/(1 - x))*exp(Sum_{k>=1} n^k*x^k/(k*(1 - x^k))).
a(n) = Sum_{j=0..n} A246935(j,n).
a(n) ~ n^n. - Vaclav Kotesovec, May 04 2018

A304782 a(n) = [x^n] (1/(1 - x))Product_{k>=1} (1 + nx^k).

Original entry on oeis.org

1, 2, 5, 19, 49, 126, 469, 1177, 2881, 6481, 23101, 53725, 127153, 274288, 581925, 1860751, 4155649, 9279791, 19409221, 39839239, 77052401, 229393207, 481747949, 1035561408, 2082441025, 4153434376, 7822058869, 14686515649, 39394280689, 79657493191, 163600884901

Offset: 0

Ilya Gutkovskiy, May 18 2018

nonn

Index entries for sequences related to partitions

Cf. A286957, A291698, A303070, A303071, A303914.

Table[SeriesCoefficient[1/(1 - x) Product[(1 + n x^k), {k, 1, n}], {x, 0, n}], {n, 0, 30}]
Table[SeriesCoefficient[1/(1 - x) Exp[Sum[(-1)^(k + 1) n^k x^k/(k (1 - x^k)), {k, 1, n}]], {x, 0, n}], {n, 0, 30}]
Table[SeriesCoefficient[QPochhammer[-n, x]/((1 + n) (1 - x)), {x, 0, n}], {n, 0, 30}]

a(n) = [x^n] (1/(1 - x))*exp(Sum_{k>=1} (-1)^(k+1)*n^k*x^k/(k*(1 - x^k))).

a(n) = Sum_{j=0..n} A286957(j,n).

cp's OEIS Frontend

A270915 Decimal expansion of a constant related to the asymptotics of A008485.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A210764 Square array T(n,k), n>=0, k>=0, read by antidiagonals in which column k gives the partial sums of column k of A144064.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A303071 a(n) = [x^n] (1/(1 - x))*Product_{k>=1} (1 + x^k)^n.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A303914 a(n) = [x^n] (1/(1 - x))*Product_{k>=1} 1/(1 - n*x^k).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A304782 a(n) = [x^n] (1/(1 - x))*Product_{k>=1} (1 + n*x^k).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A303914 a(n) = [x^n] (1/(1 - x))Product_{k>=1} 1/(1 - nx^k).

A304782 a(n) = [x^n] (1/(1 - x))Product_{k>=1} (1 + nx^k).