A134400 -id:A134400 - cp's OEIS frontend

A134401 Row sums of triangle A134400.

1, 2, 8, 24, 64, 160, 384, 896, 2048, 4608, 10240, 22528, 49152, 106496, 229376, 491520, 1048576, 2228224, 4718592, 9961472, 20971520, 44040192, 92274688, 192937984, 402653184, 838860800, 1744830464, 3623878656, 7516192768

Offset: 0

Gary W. Adamson, Oct 23 2007

Essentially the same sequence as A036289.
An elephant sequence, see A175654. For the corner squares four A[5] vectors, with decimal values 187, 190, 250 and 442, lead to this sequence. For the central square these vectors lead to the companion sequence 2*A001792, for n >= 1 and a(0)=1. - Johannes W. Meijer, Aug 15 2010
Number of vertices on a partially truncated n-cube (column 1 of A271316). - Vincent J. Matsko, Apr 07 2016

			a(3) = 24 = sum of row 3 terms of triangle A134400: (3 + 9 + 9 + 3).
a(3) = 24 = (1, 3, 3, 1) dot (1, 1, 5, 5) = (1 + 3 + 15 + 5).

Muniru A Asiru, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-4).

Cf. A001792, A036289, A097064, A134400, A175654, A271316.

a:=Concatenation([1],List([1..30],n->n*2^n)); # Muniru A Asiru, Oct 28 2018

1,seq(n*2^n,n=1..30); # Muniru A Asiru, Oct 28 2018

F = Function[x, x*2^x];F[Range[1, 10]] (* Eugeny Yakimovitch (Eugeny.Yakimovitch(AT)gmail.com), Jan 08 2008 *)
{1}~Join~Table[n 2^n, {n, 28}] (* or *) Total /@ Join[{{1}}, Table[n Binomial[n, k], {n, 28}, {k, 0, n}]] (* Michael De Vlieger, Apr 07 2016 *)

x='x+O('x^99); Vec((1-2*x+4*x^2)/(1-2*x)^2) \\ Altug Alkan, Apr 07 2016

Binomial transform of repeats of (4n+1): [1, 1, 5, 5, 9, 9, 13, 13, ...].
a(n) = n*2^n, n > 1. - Eugeny Yakimovitch (Eugeny.Yakimovitch(AT)gmail.com), Jan 08 2008
From Colin Barker, Jul 29 2012: (Start)
a(n) = 4*a(n-1) - 4*a(n-2) for n > 2.
G.f.: (1 - 2*x + 4*x^2)/(1-2*x)^2. (End)
E.g.f.: 1-E(0) where E(k)=1 - (k+1)/(1 - 2*x/(2*x - (k+1)^2/E(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Dec 07 2012
a(n) = A097064(n+1) for n >= 1. - Georg Fischer, Oct 28 2018
E.g.f.: 1 + 2*exp(2*x)*x. - Stefano Spezia, Feb 12 2023

More terms from Johannes W. Meijer, Aug 15 2010

A167930 Number of partitions of n in which some but not all parts are equal.

Original entry on oeis.org

0, 0, 0, 0, 1, 3, 4, 9, 13, 20, 29, 43, 57, 82, 110, 146, 195, 258, 334, 435, 558, 713, 910, 1150, 1446, 1814, 2268, 2815, 3491, 4308, 5301, 6501, 7954, 9692, 11795, 14295, 17301, 20876, 25148, 30200, 36218, 43322, 51741, 61650, 73354

Offset: 0

Omar E. Pol, Nov 15 2009

nonn

The parts may not all be equal, and at least one part must occur at least twice. - N. J. A. Sloane, May 30 2024

			The partitions of 6 are:
6 ....................... All parts are distinct.
5 + 1 ................... All parts are distinct.
4 + 2 ................... All parts are distinct.
4 + 1 + 1 ............... Only some parts are equal ...... (1).
3 + 3 ................... All parts are equal.
3 + 2 + 1 ............... All parts are distinct.
3 + 1 + 1 + 1 ........... Only some parts are equal ...... (2).
2 + 2 + 2 ............... All parts are equal.
2 + 2 + 1 + 1 ........... Only some parts are equal ...... (3).
2 + 1 + 1 + 1 + 1 ....... Only some parts are equal ...... (4).
1 + 1 + 1 + 1 + 1 + 1 ... All parts are equal.
Then a(6) = 4.
a(7) = 9 from 511  4111  331  322  3211  31111  2221  22111  211111. - _N. J. A. Sloane_, May 30 2024

Omar E. Pol, Illustration of the shell model of partitions (2D and 3D view)
Omar E. Pol, Illustration of the shell model of partitions (2D view)
Omar E. Pol, Illustration of the shell model of partitions (3D view)

Cf. A000005, A000009, A000041, A000065, A032741, A047967, A111133, A134400, A135010, A138121, A167931, A167932, A167933.

f[lst_]:=With[{c=Split[lst]},Length[lst]>2&&Max[Length/@c]>1&&Length[c]>1]; Table[Length[ Select[ IntegerPartitions[n],f]],{n,0,50}] (* Harvey P. Dale, May 30 2024 *)

a(n) = A047967(n) - A032741(n).
a(n) = A000041(n) - A000009(n) - A032741(n).
a(0) = 0: For n>0, a(n) = A000041(n) - A000009(n) - A000005(n) + 1.

Edited by Omar E. Pol, Nov 16 2009

More terms from Max Alekseyev, May 02 2011

A167932 Number of partitions of n such that all parts are equal or all parts are distinct.

Original entry on oeis.org

1, 1, 2, 3, 4, 4, 7, 6, 9, 10, 13, 13, 20, 19, 25, 30, 36, 39, 51, 55, 69, 79, 92, 105, 129, 144, 168, 195, 227, 257, 303, 341, 395, 451, 515, 588, 676, 761, 867, 985, 1120, 1261, 1433, 1611, 1821, 2053, 2307, 2591, 2919, 3266, 3663, 4100, 4587, 5121, 5725, 6381

Offset: 0

Omar E. Pol, Nov 15 2009

nonn

Note that for positive integers the number of partitions of n such that all parts are equal is equal to the number of proper divisors of n. (A032741(n)).

			The partitions of 6 are:
6 .............. All parts are distinct ..... (1).
5+1 ............ All parts are distinct ..... (2).
4+2 ............ All parts are distinct ..... (3).
4+1+1 .......... Only some parts are equal.
3+3 ............ All parts are equal ........ (4).
3+2+1 .......... All parts are distinct ..... (5).
3+1+1+1 ........ Only some parts are equal.
2+2+2 .......... All parts are equal ........ (6).
2+2+1+1 ........ Only some parts are equal.
2+1+1+1+1 ...... Only some parts are equal.
1+1+1+1+1+1 .... All parts are equal ........ (7).
So a(6) = 7.

Omar E. Pol, Illustration of the shell model of partitions (2D and 3D view)
Omar E. Pol, Illustration of the shell model of partitions (2D view)
Omar E. Pol, Illustration of the shell model of partitions (3D view)

Cf. A000005, A000009, A000041, A000065, A032741, A047967, A111133, A134400, A135010, A138121, A167930, A167931, A167933.

ds[n_]:=Module[{lun=Length[Union[n]]},Length[n]==lun||lun==1]; Table[ Count[ IntegerPartitions[n],?(ds)],{n,0,60}] (* _Harvey P. Dale, Sep 13 2011 *)

a(n) = A000041(n) - A167930(n).

a(n) = A000009(n) + A032741(n).

More terms from D. S. McNeil, May 10 2010

A237765 Triangular array read by rows: T(n,k) = binomial(n,2)*binomial(n,k), n>=0, 0<=k<=n.

Original entry on oeis.org

0, 0, 0, 1, 2, 1, 3, 9, 9, 3, 6, 24, 36, 24, 6, 10, 50, 100, 100, 50, 10, 15, 90, 225, 300, 225, 90, 15, 21, 147, 441, 735, 735, 441, 147, 21, 28, 224, 784, 1568, 1960, 1568, 784, 224, 28, 36, 324, 1296, 3024, 4536, 4536, 3024, 1296, 324, 36

Offset: 0

Geoffrey Critzer, Feb 12 2014

T(n,k) is the number of ways to underline exactly two elements of {1,2,...,n} and then circle exactly k elements. (The k elements that are circled are not necessarily different from the two underlined elements).
T(n,0) = T(n,n) = binomial(n,2) = A000217(n-1).
Row sums = 2^n*binomial(n,2) = A100381(n).

			0;
0,  0;
1,  2,   1;
3,  9,   9,    3;
6,  24,  36,   24,   6;
10, 50,  100,  100,  50,   10;
15, 90,  225,  300,  225,  90,   15;
21, 147, 441,  735,  735,  441,  147,  21;
28, 224, 784,  1568, 1960, 1568, 784,  224,  28;
36, 324, 1296, 3024, 4536, 4536, 3024, 1296, 324, 36;

J. Riordan, Introduction to Combinatorial Analysis, Wiley, 1958, page 14, problem #2.

Cf. A134400.

Table[Table[Binomial[n,2](Binomial[n-2,r]+2Binomial[n-2,r-1]+Binomial[n-2,r-2]),{r,0,n}],{n,0,9}]//Grid

E.g.f.: (x^2/2! + 2*y*x^2/2! + y^2*x^2/2!)*exp(y*x)*exp(x).
E.g.f. for column k: x^2/2!*exp(x)*(x^k/k! + 2*x^(k-1)/(k-1)! + x^(k-2)/(k-2)!).
T(n,k) = C(n,2)*( C(n-2,k) + 2*C(n-2,k-1) + C(n-2,k-2) ).

cp's OEIS Frontend

A134401 Row sums of triangle A134400.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Maple

Mathematica

PARI

Formula

Extensions

A167930 Number of partitions of n in which some but not all parts are equal.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A167932 Number of partitions of n such that all parts are equal or all parts are distinct.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A237765 Triangular array read by rows: T(n,k) = binomial(n,2)*binomial(n,k), n>=0, 0<=k<=n.

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Programs

Mathematica

Formula