Tilman Neumann - cp's OEIS frontend

User: Tilman Neumann

Tilman Neumann has authored 8 sequences.

A164961 Triangle read by rows: T[n,m] = quadruple factorials A001813(n) * binomials A007318(n,m).

1, 2, 2, 12, 24, 12, 120, 360, 360, 120, 1680, 6720, 10080, 6720, 1680, 30240, 151200, 302400, 302400, 151200, 30240, 665280, 3991680, 9979200, 13305600, 9979200, 3991680, 665280, 17297280, 121080960, 363242880, 605404800, 605404800

Offset: 0

Tilman Neumann, Sep 02 2009

Row sums give A052714. - Tilman Neumann, Sep 07 2009

Triangle T(n,k), read by rows, given by (2, 4, 6, 8, 10, 12, 14, ...) DELTA (2, 4, 6, 8, 10, 12, 14, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Jan 07 2012

			Triangle begins:
  1
  2, 2
  12, 24, 12
  120, 360, 360, 120
  1680, 6720, 10080, 6720, 1680

More terms. - _Tilman Neumann_, Sep 07 2009

Cf. A001813, A007318, A052714 (row sums), A084938, A085881.

T(n,k) = A085881(n,k)*2^n. - Philippe Deléham, Jan 07 2012
Recurrence equation: T(n+1,k) = (4*n+2)*(T(n,k) + T(n,k-1)). - Peter Bala, Jul 15 2012
E.g.f.: 1/sqrt(1-4*x-4*x*y). - Peter Bala, Jul 15 2012

A145201 Triangle read by rows: T(n,k) = S(n,k) mod n, where S(n,k) = Stirling numbers of the first kind.

Original entry on oeis.org

0, 1, 1, 2, 0, 1, 2, 3, 2, 1, 4, 0, 0, 0, 1, 0, 4, 3, 1, 3, 1, 6, 0, 0, 0, 0, 0, 1, 0, 4, 4, 1, 0, 2, 4, 1, 0, 0, 8, 0, 3, 0, 6, 0, 1, 0, 6, 0, 0, 5, 3, 0, 0, 5, 1, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 4, 6, 11, 6, 3, 6, 5, 6, 1, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 8, 0, 0, 0, 0, 7, 5, 7, 7, 7, 7, 7

Offset: 1

Tilman Neumann, Oct 04 2008, Oct 06 2008

The triangle T(n,k) contains many zeros. The distribution of nonzero entries is quite chaotic, but shows regular patterns, too, e.g.:
1) T(n,1) > 0 for n prime or n=4; T(n,1)=0 else
2) T(5k,k) > 0 for all k
More generally, it seems that:
3) T(pk,k) > 0 for k>0 and primes p
The following table depicts the zero (-) and nonzero (x) entries for the first 80 rows of the triangle:
-
xx
x-x
xxxx
x---x
-xxxxx
x-----x
-xxx-xxx
--x-x-x-x
-x--xx--xx
x---------x
---xxxxxxxxx
x-----------x
-x----xxxxxxxx
--x-x-x-x-x-x-x
-----xxx-x-x-xxx
x---------------x
-----x-xxx-x-x-xxx
x-----------------x
---x---xxxxx-x-xxxxx
--x---x-x---x-x---x-x
-x--------xxxx----xxxx
x---------------------x
-------x-xxx-xxx-xxx-xxx
----x---x---x---x---x---x
-x----------xx--xx--xx--xx
--------x-x-x-x-x-x-x-x-x-x
---x-----x--xxxxxxxxxxxxxxxx
x---------------------------x
-----x---x-x--xxxxxxxxxxxxxxxx
x-----------------------------x
-------------xxx-x-x-x-x-x-x-xxx
--x-------x-x-x-------x-----x-x-x
-x--------------xx--------------xx
----x-x---x---x-x-----x---x-x-x---x
-----------x-x-xxxxx---x-x-x-x-xxxxx
x-----------------------------------x
-x----------------xxxx------------xxxx
--x---------x-x---x-x-----x---x-x---x-x
-------x---x---x-xxx-xxx---x-x-x-xxx-xxx
x---------------------------------------x
-----x-----x-x-x-x-xxx-xxx---x-x-x-xxx-xxx
x-----------------------------------------x
---x---------x------xxxxxxxx-x-x-x-xxxxxxxxx
--------x---x-x-x-x-x-x-x-x-x---x-x-x-x-x-x-x
-x--------------------xxxxxxxx--------xxxxxxxx
x---------------------------------------------x
---------------x-x---xxx-x-x-xxx-x-x--xx-x-x-xxx
------x-----x-----x-----x-----x-----x-----x-----x
---------x---x---x---x--xx---x--xx---x--xx---x--xx
--x-------------x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x
---x-----------x--------xxxx-x-xxxxx---xxxxx-x-xxxxx
x---------------------------------------------------x
-----------------x-x-x-x-xxxxx-x-xxxxx-x-xxxxx-x-xxxxx
----x-----x---x---------x-----x---x---------x-----x---x
-------x-----x-----------xxx-xxx--xx-xxx-xxx-xxx-xxx-xxx
--x---------------x-x---------------x-x---------------x-x
-x--------------------------xx--xx--xx--xx--xx--xx--xx--xx
x---------------------------------------------------------x
-----------x---x---x-x-x----xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
x-----------------------------------------------------------x
-x----------------------------xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
--------x-----x-----x-x-x-x-----x-----x-x-x-x-----x-----x-x-x-x
-----------------------------xxx-x-x-x-x-x-x-x-x-x-x-x-x-x-x-xxx
----x-------x---x---x---x---x---x---x---x---x-------x---x---x---x
-----x---------x-----x-x-x-x-x--xx-x---x-x---x-x-------x-x-x---xxx
x-----------------------------------------------------------------x
---x---------------x------------xxxx-------------x-x------------xxxx
--x-------------------x-x-x-x-x-x-------x-x-x-x-x-x-------x-x-x-x-x-x
---------x---x-x-x---x---x-x-x---xxxxx---x---x---x-x-x---x---x-x-xxxxx
x---------------------------------------------------------------------x
-----------------------x-x-x-x-x-xxx-xxx-x-x-x-x-x-x-x-x---x-x-x-xxx-xxx
x-----------------------------------------------------------------------x
-x----------------------------------xx--xx--------------------------xx--xx
--------------x---x---x-x-x---x-x-x-x-x---x-x-x-x-x---x-x-x-x-x---x-x-x-x-x
---x-----------------x--------------xxxxxxxx---------x-x-x-x--------xxxxxxxx
------x---x-----x-----x---x-x-----x-x---------x-----x---x-x-----x-x---x-----x
-----x-----------x-------x-x-x-x-x-x-xxxxxxxxx-x-x-x-x-x-x-x-x-x-x-x-xxxxxxxxx
x-----------------------------------------------------------------------------x
---------------x---x---------------x-xxx-x-x-xxx---x---x-x-x-x-x---x-xxx-x-x-xxx
SUM(A057427(a(k)): 1<=k<=n) = A005127(n). - Reinhard Zumkeller, Jul 04 2009

			Triangle starts:
0;
1, 1;
2, 0, 1;
2, 3, 2, 1;
4, 0, 0, 0, 1;
0, 4, 3, 1, 3, 1;
6, 0, 0, 0, 0, 0, 1;
....

Cf. A000040, A008275, A061006 (first column).

tabl(nn) = {for (n=1, nn, for (k=1, n, print1(stirling(n, k, 1) % n, ", ");); print(););} \\ Michel Marcus, Aug 10 2015

T(n,k) = S(n,k) mod n, where S(n,k) = Stirling numbers of the first kind.

A145520 Triangle read by rows: T2[n,k] = Sum_{partitions of n with k parts p(n, k; m_1, m_2, m_3, ..., m_n)} c(n; m_1, m_2, ..., m_n) * x_1^m_1 * x_2^m_2 * ... x^n*m_n, where x_i = i-th prime.

Original entry on oeis.org

2, 3, 4, 5, 18, 8, 7, 67, 72, 16, 11, 220, 470, 240, 32, 13, 697, 2625, 2420, 720, 64, 17, 2100, 13559, 20230, 10360, 2016, 128, 19, 6159, 66374, 152313, 120400, 39200, 5376, 256, 23, 17340, 313136, 1071168, 1235346, 602784, 135744, 13824, 512, 29, 47581

Offset: 1

Tilman Neumann, Oct 12 2008, Oct 13 2008, Sep 02 2009

Here c(n; m_1, m_2, ..., m_n) = n! / (m_1!*1!^m_1 * m_2!*2!^m_2 * ... * m_n!*n!^m_n) is the number of ways to realize the partition p(n, k; m_1, m_2, m_3, ..., m_n).

Also the Bell transform of the prime numbers. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 29 2016

			Triangle begins:
:  2;
:  3,    4;
:  5,   18,     8;
:  7,   67,    72,    16;
: 11,  220,   470,   240,    32;
: 13,  697,  2625,  2420,   720,   64;
: 17, 2100, 13559, 20230, 10360, 2016, 128;

Alois P. Heinz, Rows n = 1..141, flattened
Tilman Neumann, More terms, partitions generator and transform implementation

Cf. A000040, A007446 (row sums), A145518.

b:= proc(n) option remember; expand(`if`(n=0, 1, add(x
      *binomial(n-1, j-1)*ithprime(j)*b(n-j), j=1..n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n)):
seq(T(n), n=1..10);  # Alois P. Heinz, May 27 2015
# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> ithprime(n+1), 9); # Peter Luschny, Jan 29 2016

b[n_] := b[n] = Expand[If[n == 0, 1, Sum[x*Binomial[n - 1, j - 1]*Prime[j]* b[n - j], {j, 1, n}]]]; T[n_] := Function [p, Table[Coefficient[p, x, i], {i, 1, n}]][b[n]]; Table[T[n], {n, 1, 10}] // Flatten (* Jean-François Alcover, Jan 23 2016, after Alois P. Heinz *)
BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 12;
B = BellMatrix[Function[n, Prime[n+1]], rows];
Table[B[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* Jean-François Alcover, Jun 28 2018, after Peter Luschny *)

A145518 Triangle read by rows: T1[n,k;x] := Sum_{partitions with k parts p(n, k; m_1, m_2, m_3, ..., m_n)} x_1^m_1 * x_2^m_2 * ... x^n*m_n, for x_i = A000040(i).

Original entry on oeis.org

2, 3, 4, 5, 6, 8, 7, 19, 12, 16, 11, 29, 38, 24, 32, 13, 68, 85, 76, 48, 64, 17, 94, 181, 170, 152, 96, 128, 19, 177, 326, 443, 340, 304, 192, 256, 23, 231, 683, 787, 886, 680, 608, 384, 512, 29, 400, 1066, 1780, 1817, 1772, 1360, 1216, 768, 1024, 31, 484, 1899, 3119

Offset: 1

Tilman Neumann, Oct 12 2008

Let p(n; m_1, m_2, m_3, ..., m_n) denote a partition of integer n in exponential representation, i.e., the m_i are the counts of parts i and satisfy 1*m_1 + 2*m_2 + 3*m_3 + ... + n*m_n = n.
Let p(n, k; m_1, m_2, m_3, ..., m_n) be the partitions of n into exactly k parts; these are further constrained by m_1 + m_2 + m_3 + ... + m_n = k.
Then the triangle is given by T1[n,k;x] := Sum_{all p(n, k; m_1, m_2, m_3, ..., m_n)} x_1^m_1 * x_2^m_2 * ... x^n*m_n, where x_i is the i-th prime number (A000040).
2nd column (4, 6, 19, 29, 68, 94, 177, ...) is A024697.
Row sums give A145519.

			Triangle starts:
   2;
   3,   4;
   5,   6,   8;
   7,  19,  12,  16;
  11,  29,  38,  24,  32;
  13,  68,  85,  76,  48,  64;
  ...

Alois P. Heinz, Rows n = 1..141, flattened
Tilman Neumann, More terms, partition generator and transform implementation.

Cf. A000040, A024697, A145519, A145520, A258323.

g:= proc(n, i) option remember; `if`(n=0 or i=1, (2*x)^n,
      expand(add(g(n-i*j, i-1)*(ithprime(i)*x)^j, j=0..n/i)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(g(n$2)):
seq(T(n), n=1..12);  # Alois P. Heinz, May 25 2015

g[n_, i_] := g[n, i] = If[n==0 || i==1, (2 x)^n, Expand[Sum[g[n-i*j, i-1]*(Prime[i]*x)^j, {j, 0, n/i}]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 1, n}]][g[n, n]]; Table[T[n], {n, 1, 12}] // Flatten (* Jean-François Alcover, Jul 15 2015, after Alois P. Heinz *)

Reference to more terms etc. changed to make it version independent by Tilman Neumann, Sep 02 2009

A145177 Denominators of rational coefficients in series expansion of 1/(Bernoulli trial entropy).

Original entry on oeis.org

2, 6, 4, 12, 6, 8, 20, 9, 8, 16, 30, 90, 48, 12, 32, 42, 720, 2160, 12, 96, 64, 56, 2520, 1440, 540, 576, 32, 128, 72, 25200, 10080, 2592, 1728, 24, 384, 256, 90, 700, 302400, 22680, 5184, 4320, 256, 96, 512, 110, 75600, 6720, 21600, 108864, 34560, 34560, 288

Offset: 1

Tilman Neumann, Oct 03 2008, Oct 04 2008

This triangle T[n,k] is given by the denominators of rational coefficients R[n,k] appearing in a certain series expansion of 1/S(x) around x=0,
where S(x) = -x*log(x) - (1-x)*log(1-x) is the Bernoulli trial entropy.
The series is
1/S(x) = 1/(x*(1-log(x))) + sum_{n=1..inf} x^(n-1) * sum_{k=1..n} R[n,k]/(1-log(x))^(k+1)
= 1/(x*(1-log(x))) * (1 + sum_{n=1..inf} x^n * sum_{k=1..n} R[n,k]/(1-log(x))^k).
The first rationals R[n,k] are
1/2
1/6     1/4
1/12    1/6       1/8
1/20    1/9       1/8      1/16
1/30    7/90      5/48     1/12    1/32
1/42   41/720   181/2160   1/12    5/96   1/64
1/56  109/2520   97/1440  41/540  35/576  1/32  1/128
The LCM of the rows of T[n,k], i.e., A003418(A145177(n,1), ..., A145177(n,n)), is just A091137(n).
See A145176 for the numerators of R[n,k] and A145178 for the numerators scaled to denominators A091137.

Robert Israel, Table of n, a(n) for n = 1..10011(first 141 rows, flattened)

Cf. A003418, A091137, A145176, A145178.

f:= -x*log(x)-(1-x)*log(1-x):
S:= map(normal,eval(series(x*(1-ln(x))/f, x, 12),ln(x)=1-1/t)):
for n from 1 to 10 do
  C:= coeff(S,x,n);
  for k from 1 to n do T[n,k]:= denom(coeff(C,t,k)) od
od:
seq(seq(T[n,k],k=1..n),n=1..10); # Robert Israel, Jul 09 2015

ORDER:=14: expand(_invert(series(-x*ln(x)-(1-x)*ln(1-x), x=0)));

A145176 Numerators of coefficients in series expansion of 1/(Bernoulli trial entropy).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 5, 1, 1, 1, 41, 181, 1, 5, 1, 1, 109, 97, 41, 35, 1, 1, 1, 853, 551, 173, 107, 1, 7, 1, 1, 19, 13579, 1313, 307, 203, 7, 1, 1, 1, 1679, 251, 1081, 5969, 1681, 1169, 5, 3, 1, 1, 1537, 3169, 4913, 13583, 3481, 7819, 101, 11, 5, 1, 1, 18167

Offset: 1

Tilman Neumann, Oct 03 2008, Oct 04 2008

This triangle T[n,k] is given by the numerators of rational coefficients R[n,k] appearing in a certain series expansion of 1/S(x) around x0=0,
where S(x) = - x*log(x) - (1-x)*log(1-x) is the Bernoulli trial entropy.
The series is
1/S(x) = 1/(x*(1-log(x))) + sum_{n=1..inf} x^(n-1) * sum_{k=1..n} R[n,k]/(1-log(x))^(k+1)
= 1/(x*(1-log(x))) * (1 + sum_{n=1..inf} x^n * sum_{k=1..n} R[n,k]/(1-log(x))^k)
The first rationals R[n,k] are
1/2
1/6 1/4
1/12 1/6 1/8
1/20 1/9 1/8 1/16
1/30 7/90 5/48 1/12 1/32
1/42 41/720 181/2160 1/12 5/96 1/64
1/56 109/2520 97/1440 41/540 35/576 1/32 1/128
See A145177 for the denominators of R[n,k] and A145178 for numerators scaled to denominators given by A091137.

Robert Israel, Table of n, a(n) for n = 1..10011 (first 141 rows, flattened).

Cf. A145177, A145178, A091137.

f:= -x*log(x)-(1-x)*log(1-x):
S:= map(normal,eval(series(x*(1-ln(x))/f, x, 12),ln(x)=1-1/t)):
for n from 1 to 141 do
  C:= coeff(S,x,n);
  for k from 1 to n do T[n,k]:= numer(coeff(C,t,k));
 od
od:
seq(seq(T[n,k],k=1..n),n=1..10); # Robert Israel, Jul 09 2015

ORDER:=14: expand(_invert(series(-x*ln(x)-(1-x)*ln(1-x), x=0)));

A145519 a(n) = Sum_{k=1..n} A145518(n,k).

Original entry on oeis.org

1, 2, 7, 19, 54, 134, 354, 838, 2057, 4794, 11232, 25412, 58075, 128670, 286152, 625829, 1365653, 2941088, 6331146, 13474533, 28642325, 60404681, 127082128, 265712673, 554608226, 1151374963, 2385950536, 4924685252, 10145267212, 20831428273, 42708248451

Offset: 0

Tilman Neumann, Oct 12 2008

nonn

Row sums of A145518.
Also row sums of A129129, A215366.
a(n) = sum of the Heinz numbers of the partitions of n. The Heinz number of a partition p = [p_1, p_2, ..., p_r] is defined as Product(p_j-th prime, j=1...r) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). For example, for the 3 partitions of 3, namely [3], [1,2], and [1,1,1] we get 5, 2*3=6, and 2*2*2=8, respectively; their sum is a(3) = 19. - Emeric Deutsch, Jun 09 2015

Alois P. Heinz, Table of n, a(n) for n = 0..1000
More terms in A145518 and A145519

Cf. A129129, A145518, A215366, A334434.

b:= proc(n, i) option remember; `if`(n=0 or i<2, 2^n,
      add(b(n-i*j, i-1)*ithprime(i)^j, j=0..iquo(n, i)))
    end:
a:= n-> b(n, n):
seq(a(n), n=0..35);  # Alois P. Heinz, Feb 19 2013

b[n_, i_] := b[n, i] = If[n == 0 || i < 2, 2^n, Sum[b[n-i*j, i-1]*Prime[i]^j, {j, 0, Quotient[n, i]}]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Feb 24 2015, after Alois P. Heinz *)

G.f.: 1/Product_{i>=1}(1-prime(i)*x^i). - Vladeta Jovovic, Nov 09 2008

a(n) ~ c * 2^n, where c = Product_{k>=2} 1/(1 - prime(k)/2^k) = 50.412394245500690832088704444961002125578414895935257436317... . - Vaclav Kotesovec, Sep 10 2014, updated Apr 11 2020

a(0) inserted by Alois P. Heinz, Feb 19 2013

A145178 Numerators of coefficients of series expansion of 1/(Bernoulli trial entropy), scaled to denominators A091137.

Original entry on oeis.org

1, 2, 3, 2, 4, 3, 36, 80, 90, 45, 48, 112, 150, 120, 45, 1440, 3444, 5068, 5040, 3150, 945, 2160, 5232, 8148, 9184, 7350, 3780, 945, 50400, 122832, 198360, 242200, 224700, 151200, 66150, 14175, 80640, 196992, 325896, 420160, 429800, 341040, 198450

Offset: 1

Tilman Neumann, Oct 03 2008

These are the numerators A145176 scaled to denominators A091137.

In other words: A145178(n,k)/A091137(n) = A145176(n,k)/A145177(n,k)

Cf. A091137, A145176, A145177

Corrected description of the series. - Tilman Neumann, Oct 04 2008

cp's OEIS Frontend

User: Tilman Neumann

A164961 Triangle read by rows: T[n,m] = quadruple factorials A001813(n) * binomials A007318(n,m).

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A145201 Triangle read by rows: T(n,k) = S(n,k) mod n, where S(n,k) = Stirling numbers of the first kind.

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A145520 Triangle read by rows: T2[n,k] = Sum_{partitions of n with k parts p(n, k; m_1, m_2, m_3, ..., m_n)} c(n; m_1, m_2, ..., m_n) * x_1^m_1 * x_2^m_2 * ... x^n*m_n, where x_i = i-th prime.

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Maple

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A145518 Triangle read by rows: T1[n,k;x] := Sum_{partitions with k parts p(n, k; m_1, m_2, m_3, ..., m_n)} x_1^m_1 * x_2^m_2 * ... x^n*m_n, for x_i = A000040(i).

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Maple

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A145177 Denominators of rational coefficients in series expansion of 1/(Bernoulli trial entropy).

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Maple

MuPAD

A145176 Numerators of coefficients in series expansion of 1/(Bernoulli trial entropy).

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Maple

MuPAD

A145519 a(n) = Sum_{k=1..n} A145518(n,k).

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A145178 Numerators of coefficients of series expansion of 1/(Bernoulli trial entropy), scaled to denominators A091137.

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