A000713 -id:A000713 - cp's OEIS frontend

A210843 Level of the n-th plateau of the column k of the square array A195825, when k -> infinity.

1, 4, 13, 35, 86, 194, 415, 844, 1654, 3133, 5773, 10372, 18240, 31449, 53292, 88873, 146095, 236977, 379746, 601656, 943305, 1464501, 2252961, 3436182, 5198644, 7805248, 11634685, 17224795, 25336141, 37038139, 53828275, 77792869

Offset: 1

Omar E. Pol, Jun 19 2012

nonn

Also the first (k+1)/2 terms of this sequence are the levels of the (k+1)/2 plateaus of the column k of A195825, whose lengths are k+1, k-1, k-3, k-5,... 2, if k is odd.
Also the first k/2 terms of this sequence are the levels of the k/2 plateaus of the column k of A195825, whose lengths are k+1, k-1, k-3, k-5,... 3, if k is a positive even number.
For the visualization of the plateaus see the graph of the sequences mentioned in crossrefs section (columns k=1..10 of A195825), for example see the graph of A210964.
Also numbers that are repeated in column k of square array A195825, when k -> infinity.
Note that the definition and the comments related to the square array A195825 mentioned above are also valid for the square array A211970, since both arrays contains the same columns, if k >= 1.
Is this the EULER transform of 4, 3, 3, 3, 3, 3, 3...?

			Column 1 of A195825 is A000041 which starts: [1, 1], 2, 3, 5, 7, 11... The column contains only one plateau: [1, 1] which has level 1 and length 2. So a(1) = 1.
Column 3 of A195825 is A036820 which starts: [1, 1, 1, 1], 2, 3, [4, 4], 5, 7, 10... The column contains only two plateaus: [1, 1, 1, 1], [4, 4], which have levels 1, 4 and lengths 4, 2. So a(1)= 1 and a(2) = 2.
Column 6 of A195825 is A195850 which starts: [1, 1, 1, 1, 1, 1, 1], 2, 3, [4, 4, 4, 4, 4], 5, 7, 10, 12, [13, 13, 13], 14, 16, 21... The column contains three plateaus: [1, 1, 1, 1, 1, 1, 1], [4, 4, 4, 4, 4], [13, 13, 13], which have levels 1, 4, 13 and lengths 7, 5, 3. So a(1) = 1, a(2) = 4 and a(3) = 13.

Partial sums of A000716. Column 3 of A210764.
Columns (k=1..10) of A195825: A000041, A006950, A036820, A195848, A195849, A195850, A195851, A195852, A196933, A210964.
Cf. A000070, A000712, A000713, A010815, A211970.

CoefficientList[Series[1/(1-x)*Product[1/(1-x^k)^3,{k,1,50}],{x,0,50}],x] (* Vaclav Kotesovec, Aug 16 2015 *)

From Vaclav Kotesovec, Aug 16 2015: (Start)
a(n) ~ sqrt(2*n)/Pi * A000716(n).
a(n) ~ exp(sqrt(2*n)*Pi) / (8*Pi*n).
(End)

A303070 a(n) = [x^n] (1/(1 - x))*Product_{k>=1} 1/(1 - x^k)^n.

Original entry on oeis.org

1, 2, 8, 35, 164, 787, 3857, 19147, 96004, 485009, 2465013, 12589315, 64555985, 332158127, 1714001409, 8866730665, 45968787524, 238778897128, 1242417984179, 6474394344503, 33784931507529, 176515163156311, 923265560495737, 4834081924982522, 25334170138318345, 132883719945537587

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

Main diagonal of A210764.

Cf. A000070, A000713, A008485, A144064, A210843.

Table[SeriesCoefficient[1/(1 - x) Product[1/(1 - x^k)^n, {k, 1, n}], {x, 0, n}], {n, 0, 25}]
Table[SeriesCoefficient[1/(1 - x) Exp[n Sum[x^k/(k (1 - x^k)), {k, 1, n}]], {x, 0, n}], {n, 0, 25}]

a(n) = [x^n] (1/(1 - x))*exp(n*Sum_{k>=1} x^k/(k*(1 - x^k))).
a(n) = A210764(n,n) = Sum_{j=0..n} A144064(j,n).
a(n) ~ c * d^n / sqrt(n), where d = A270915 = 5.352701333486642687772415814165... and c = 0.4068869940800214657298372785820... - Vaclav Kotesovec, May 19 2018

A146023 Triangle read by rows, square of A027293.

Original entry on oeis.org

1, 2, 1, 5, 2, 1, 10, 5, 2, 1, 20, 10, 5, 2, 1, 36, 20, 10, 5, 2, 1, 65, 36, 20, 10, 5, 2, 1, 110, 65, 36, 20, 10, 5, 2, 1, 185, 110, 65, 36, 20, 10, 5, 2, 1, 300, 185, 110, 65, 36, 20, 10, 5, 2, 1

Offset: 0

Gary W. Adamson, Oct 26 2008

Triangle, A000712 (1, 2, 5, 10, 20, 36,...) in every column.
Row sums = A000713: (1, 3, 8, 18, 38, 74, 139,...)
Note that we are working here in the world of lower triangular matrices. - N. J. A. Sloane, Jun 15 2020

			First few rows of the triangle =
    1;
    2,  1;
    5,  2,  1;
   10,  5,  2,  1;
   20, 10,  5,  2,  1;
   36, 20, 10,  5,  2,  1;
   65, 36, 20, 10,  5,  2,  1;
  110, 65, 36, 20, 10,  5,  2,  1;
...

Robert Price, Table of n, a(n) for n = 0..819 (first 40 rows)

Cf. A000712, A000713, A027293.

lim = 10;
A000712 = Table [Length@IntegerPartitions[n, All, Range@n~Join~Range@n], {n, 0, lim - 1}]
Table[Reverse[Take[A000712, n]], {n, lim}] // Flatten (* Robert Price, Jun 15 2020 *)

Extraneous data deleted by Robert Price, Jun 15 2020

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 *)

A093010 Triangle, read by rows, such that the convolution of the n-th row with the natural numbers forms the n-th diagonal, for n>=0, where each row begins with 1.

Original entry on oeis.org

1, 1, 2, 1, 4, 3, 1, 6, 7, 4, 1, 8, 14, 10, 5, 1, 10, 22, 22, 13, 6, 1, 12, 33, 40, 30, 16, 7, 1, 14, 45, 66, 58, 38, 19, 8, 1, 16, 60, 100, 104, 76, 46, 22, 9, 1, 18, 76, 146, 168, 142, 94, 54, 25, 10, 1, 20, 95, 202, 262, 242, 180, 112, 62, 28, 11, 1, 22, 115, 272, 386, 394, 316

Offset: 0

Paul D. Hanna, Mar 14 2004

Row sums form A000713, the number of partitions of n into parts of 3 kinds. Antidiagonal sums form A000990, the number of 2-line partitions of n.

			T(7,3) = 66 = 1*4+8*3+14*2+10*1 = T(4,0)*4+T(4,1)*3+T(4,2)*2+T(4,3)*1; this is also the third term of the 4th-diagonal.
The 6th antidiagonal is {1,10,14,4}, which has a sum of 29 = A000990(6) = number of 2-line partitions of 6.
Rows begin:
{1},
{1,2},
{1,4,3},
{1,6,7,4},
{1,8,14,10,5},
{1,10,22,22,13,6},
{1,12,33,40,30,16,7},
{1,14,45,66,58,38,19,8},
{1,16,60,100,104,76,46,22,9},
{1,18,76,146,168,142,94,54,25,10},
{1,20,95,202,262,242,180,112,62,28,11},
{1,22,115,272,386,394,316,218,130,70,31,12},...

Cf. A000713, A000990, A092905.

PARI
```
T(n,k)=if(n
				
```

T(n, k) = sum_{j=0..k} (k-j+1)*T(n-k, j), with T(0, n) = 1 for all n>=0.
A000713(n) = sum_{k=0..n} T(n, k) (row sums).
A000990(n) = sum_{k=0..floor(n/2)} T(n-k, k) (antidiagonal sums).

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A210843 Level of the n-th plateau of the column k of the square array A195825, when k -> infinity.

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A303070 a(n) = [x^n] (1/(1 - x))*Product_{k>=1} 1/(1 - x^k)^n.

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A146023 Triangle read by rows, square of A027293.

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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.

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A093010 Triangle, read by rows, such that the convolution of the n-th row with the natural numbers forms the n-th diagonal, for n>=0, where each row begins with 1.

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