A227402 -id:A227402 - cp's OEIS frontend

A227413 a(1)=1, a(2n)=nthprime(a(n)), a(2n+1)=nthcomposite(a(n)), where nthprime = A000040, nthcomposite = A002808.

1, 2, 4, 3, 6, 7, 9, 5, 8, 13, 12, 17, 14, 23, 16, 11, 10, 19, 15, 41, 22, 37, 21, 59, 27, 43, 24, 83, 35, 53, 26, 31, 20, 29, 18, 67, 30, 47, 25, 179, 58, 79, 34, 157, 54, 73, 33, 277, 82, 103, 40, 191, 62, 89, 36, 431, 114, 149, 51, 241, 75, 101, 39, 127, 46

Offset: 1

Antti Karttunen, Jul 10 2013

Inverse permutation of A135141.

Shares with A073846 the property that the other bisection consists of just primes and the other bisection of just nonprimes.

Antti Karttunen and Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Index entries for sequences that are permutations of the natural numbers

Similarly constructed permutations: A227402, A227404, A227410, A227412. Cf. also A073846, A209636.

import Data.List (transpose)
a227413 n = a227413_list !! (n-1)
a227413_list = 1 : concat (transpose [map a000040 a227413_list,
                                      map a002808 a227413_list])
-- Reinhard Zumkeller, Jan 29 2014

a(1)=1, a(2n) = A000040(a(n)), a(2n+1) = A002808(a(n)).

A007097(n) = a(A000079(n)).

A226031 Number A(n,k) of unimodal functions f:[n]->[k*n]; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 4, 0, 1, 3, 16, 22, 0, 1, 4, 36, 161, 130, 0, 1, 5, 64, 525, 1716, 791, 0, 1, 6, 100, 1222, 8086, 18832, 4900, 0, 1, 7, 144, 2360, 24616, 128248, 210574, 30738, 0, 1, 8, 196, 4047, 58730, 510664, 2072862, 2385644, 194634, 0

Offset: 0

Alois P. Heinz, May 23 2013

			Square array A(n,k) begins:
  1,   1,     1,      1,      1,       1, ...
  0,   1,     2,      3,      4,       5, ...
  0,   4,    16,     36,     64,     100, ...
  0,  22,   161,    525,   1222,    2360, ...
  0, 130,  1716,   8086,  24616,   58730, ...
  0, 791, 18832, 128248, 510664, 1505205, ...

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

Columns k=0-2 give: A000007, A088536, A226012.
Rows n=0-2 give: A000012, A001477, A016742.
Main diagonal gives: A227402.
Cf. A071920.

A:= (n, k)-> `if`(n=0, 1, add(binomial(n+2*j-1, 2*j), j=0..k*n-1)):
seq(seq(A(n, d-n), n=0..d), d=0..10);

A[n_, k_] := If[n==0, 1, Sum[Binomial[n + 2j - 1, 2j], {j, 0, k n - 1}]];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Dec 20 2020, after Alois P. Heinz *)

A(n,k) = Sum_{j=0..k*n-1} C(n+2*j-1,2*j), A(0,k) = 1.

A(n,k) = A071921(n,k*n).

A227406 Number of unimodal functions f:[n]->[2^n].

Original entry on oeis.org

1, 2, 16, 372, 24616, 5014592, 3349471840, 7649590386464, 61356625102897216, 1758844330913892684288, 182379122144778004351027200, 69026760045145802122822210022400, 96048744530120196897251255933762037760, 494360393380904255996973467025921794482614272

Offset: 0

Alois P. Heinz, Sep 21 2013

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..50

Cf. A071920, A227402.

a:= n-> sum(binomial(n+2*j-1, 2*j), j=0..2^n-1):
seq(a(n), n=0..20);

Table[Sum[Binomial[n+2*j-1,2*j],{j,0,2^n-1}],{n,0,15}] (* Vaclav Kotesovec, Sep 22 2013 *)

a(n) = Sum_{j=0..2^n-1} C(n+2*j-1,2*j).
a(n) = A071921(n,2^n).
a(n) ~ 2^(n^2+n-1)/n!. - Vaclav Kotesovec, Sep 22 2013

cp's OEIS Frontend

A227413 a(1)=1, a(2n)=nthprime(a(n)), a(2n+1)=nthcomposite(a(n)), where nthprime = A000040, nthcomposite = A002808.

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A226031 Number A(n,k) of unimodal functions f:[n]->[k*n]; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A227406 Number of unimodal functions f:[n]->[2^n].

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