A189890 -id:A189890 - cp's OEIS frontend

A145515 Square array A(n,k), n>=0, k>=0, read by antidiagonals: A(n,k) is the number of partitions of k^n into powers of k.

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 4, 1, 1, 1, 2, 5, 10, 1, 1, 1, 2, 6, 23, 36, 1, 1, 1, 2, 7, 46, 239, 202, 1, 1, 1, 2, 8, 82, 1086, 5828, 1828, 1, 1, 1, 2, 9, 134, 3707, 79326, 342383, 27338, 1, 1, 1, 2, 10, 205, 10340, 642457, 18583582, 50110484, 692004, 1, 1, 1, 2, 11, 298, 24901, 3649346, 446020582, 14481808030, 18757984046, 30251722, 1, 1

Offset: 0

Alois P. Heinz, Oct 11 2008

			A(2,3) = 5, because there are 5 partitions of 3^2=9 into powers of 3: [1,1,1,1,1,1,1,1,1], [1,1,1,1,1,1,3], [1,1,1,3,3], [3,3,3], [9].
Square array A(n,k) begins:
  1,  1,   1,    1,     1,      1,  ...
  1,  1,   2,    2,     2,      2,  ...
  1,  1,   4,    5,     6,      7,  ...
  1,  1,  10,   23,    46,     82,  ...
  1,  1,  36,  239,  1086,   3707,  ...
  1,  1, 202, 5828, 79326, 642457,  ...

Alois P. Heinz, Antidiagonals n = 0..40, flattened

Columns k=0+1, 2-10 give: A000012, A002577, A078125, A078537, A111822, A111827, A111832, A111837, A145512, A145513.
Row n=3 gives: A189890(k+1).
Main diagonal gives: A145514.
Cf. A007318.

b:= proc(n, j, k) local nn;
      nn:= n+1;
      if n<0  then 0
    elif j=0  or n=0 or k<=1 then 1
    elif j=1  then nn
    elif n>=j then (nn-j) *binomial(nn, j) *add(binomial(j, h)
                   /(nn-j+h) *b(j-h-1, j, k) *(-1)^h, h=0..j-1)
              else b(n, j, k):= b(n-1, j, k) +b(k*n, j-1, k)
      fi
    end:
A:= (n, k)-> b(1, n, k):
seq(seq(A(n, d-n), n=0..d), d=0..13);

b[n_, j_, k_] := Module[{nn = n+1}, Which[n < 0, 0, j == 0 || n == 0 || k <= 1, 1, j == 1, nn, n >= j, (nn-j)*Binomial[nn, j]*Sum[Binomial[j, h]/(nn-j+h)* b[j-h-1, j, k]*(-1)^h, {h, 0, j-1}], True, b[n, j, k] = b[n-1, j, k] + b[k*n, j-1, k] ] ]; a[n_, k_] := b[1, n, k]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 13}] // Flatten (* Jean-François Alcover, Dec 12 2013, translated from Maple *)

See program.

For k>1: A(n,k) = [x^(k^n)] 1/Product_{j>=0} (1-x^(k^j)).

Edited by Alois P. Heinz, Jan 12 2011

A191821 a(n) = n(2^n - n + 1) + 2^(n-1)(n^2 - 3*n + 2).

Original entry on oeis.org

2, 6, 26, 100, 332, 994, 2774, 7368, 18872, 47014, 114578, 274300, 647012, 1507146, 3473198, 7929616, 17956592, 40369870, 90177194, 200277636, 442498652, 973078066, 2130705926, 4647288280, 10099883432, 21877489014, 47244639554, 101737037068

Offset: 1

Adeniji, Adenike, Jun 17 2011

Conjecture: generating function = -((2 (-1+6 x-19 x^2+31 x^3-22 x^4+4 x^5))/(1-3 x+2 x^2)^3) - Harvey P. Dale, May 10 2021

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (9,-33,63,-66,36,-8).

Cf. A188377, A188947, A189890.

[n*(2^n-n+1)+2^(n-1)*(n^2-3*n+2): n in [1..40]]; // Vincenzo Librandi, Nov 25 2011

LinearRecurrence[{9,-33,63,-66,36,-8},{2,6,26,100,332,994},50] (* Vincenzo Librandi, Nov 25 2011 *)
Table[n(2^n-n+1)+2^(n-1) (n^2-3n+2),{n,30}] (* Harvey P. Dale, May 10 2021 *)

a(n)=(n^2-n+2)<<(n-1)-n*(n-1) \\ Charles R Greathouse IV, Jul 13 2011

G.f.: -2*x*(-1 + 6*x - 19*x^2 + 31*x^3 - 22*x^4 + 4*x^5) / ( (2*x-1)^3*(x-1)^3 ). - R. J. Mathar, Aug 26 2011

A190531 Number of idempotents in Identity Difference Partial Transformation semigroup.

Original entry on oeis.org

2, 5, 17, 57, 185, 593, 1901, 6121, 19793, 64161, 208085, 674105, 2179001, 7023409, 22566269, 72268809, 230696609, 734153537, 2329503653, 7371475033, 23267249417, 73268609745, 230224239437, 721965697577, 2259855722225

Offset: 1

Adeniji, Adenike, Jun 04 2011

nonn

IDP_n is a semigroup with the non-isolation property and E(IDP_n) denotes the set of idempotents (satisfying e^2 = e) in IDP_n.

#E(IDP_n) is the number of idempotent elements in the semigroup IDP_n for each n in N. E(IDP_n) is a subset of partial transformation semigroup having the property that the difference in the image, Im(alpha), is not greater than 1 and e^2 = e for each e in IDP_n.

			Example: For n=4, #IDP_n = 3*9 + 4*8 - 4 + 2 = 27 + 32 - 2 = 57

Index entries for linear recurrences with constant coefficients, signature (12,-58,144,-193,132,-36).

Cf. A189890.

LinearRecurrence[{12,-58,144,-193,132,-36},{2,5,17,57,185,593},30] (* Harvey P. Dale, Apr 11 2020 *)

#IDP_n = (n-1)*3^(n-2) + n*2^(n-1) - n + 2.

G.f.: -x*(-2+19*x-73*x^2+145*x^3-153*x^4+68*x^5) / ( (x-1)^2*(3*x-1)^2*(2*x-1)^2 ). - R. J. Mathar, Jun 19 2011

cp's OEIS Frontend

A145515 Square array A(n,k), n>=0, k>=0, read by antidiagonals: A(n,k) is the number of partitions of k^n into powers of k.

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A191821 a(n) = n(2^n - n + 1) + 2^(n-1)(n^2 - 3*n + 2).

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A190531 Number of idempotents in Identity Difference Partial Transformation semigroup.

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cp's OEIS Frontend

A145515 Square array A(n,k), n>=0, k>=0, read by antidiagonals: A(n,k) is the number of partitions of k^n into powers of k.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A191821 a(n) = n*(2^n - n + 1) + 2^(n-1)*(n^2 - 3*n + 2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A190531 Number of idempotents in Identity Difference Partial Transformation semigroup.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A191821 a(n) = n(2^n - n + 1) + 2^(n-1)(n^2 - 3*n + 2).