cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 31-32 of 32 results.

A259760 Triangle read by rows: T(n,k) is the number of partial idempotent mappings (of an n-chain) with breadth exactly k.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 3, 9, 10, 1, 4, 18, 40, 41, 1, 5, 30, 100, 205, 196, 1, 6, 45, 200, 615, 1176, 1057, 1, 7, 63, 350, 1435, 4116, 7399, 6322, 1, 8, 84, 560, 2870, 10976, 29596, 50576, 41393, 1, 9, 108, 840, 5166, 24696, 88788, 227592, 372537, 293608
Offset: 0

Views

Author

Wafa AlNadabi, Jul 04 2015

Keywords

Examples

			T(3,2) = 9 because there are exactly 9 partial idempotent mappings (of a 3-chain) with breadth exactly 2, namely: (12-->11), (12-->22), (12-->12), (13-->11), (13-->33), (13-->13), (23-->22), (23-->33), (23-->23).
Triangle starts:
1;
1, 1;
1, 2, 3;
1, 3, 9, 10;
1, 4, 18, 40, 41;
...

References

  • F. AlKharosi, W. AlNadabi and A. Umar, "Combinatorial results for idempotents in full and partial transformation semigroups", (submitted).

Links

Crossrefs

Cf. A000248, A052512.
Row sums give A080108(n+1).

Programs

  • PARI
    tabl(nn) = {for (n=0, nn, for (k=0, n, print1(binomial(n,k)*sum(m=0, k, binomial(k,m)*m^(k-m)), ", ");); print(););} \\ Michel Marcus, Jul 15 2015

Formula

T(n,k) = binomial(n,k) * Sum_{m=0..k} binomial(k,m)*m^(k-m).

Extensions

More terms from Michel Marcus, Jul 15 2015

A325219 G.f. A(x) satisfies: Sum_{n>=0} x^n*A(x)^(n^2) = Sum_{n>=0} x^n/(1-n*x)^n.

Original entry on oeis.org

1, 1, 1, 3, 14, 81, 554, 4175, 33894, 292482, 2658803, 25312031, 251337905, 2595476384, 27814372541, 308814996237, 3547597450937, 42121414823717, 516406224737906, 6531681539263289, 85162992707351910, 1143744473741844428, 15809784290241899546, 224756696173450416445, 3283701348287927969258, 49267186208121297961411, 758541179347396245098635, 11976195590135148979244826, 193765334786286246261399910
Offset: 0

Views

Author

Paul D. Hanna, Apr 30 2019

Keywords

Examples

			G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 14*x^4 + 81*x^5 + 554*x^6 + 4175*x^7 + 33894*x^8 + 292482*x^9 + 2658803*x^10 + 25312031*x^11 + 251337905*x^12 + ...
such that the following series are equal
B(x) = 1 + x*A(x) + x^2*A(x)^4 + x^3*A(x)^9 + x^4*A(x)^16 + x^5*A(x)^25 + x^6*A(x)^36 + x^7*A(x)^49 + x^8*A(x)^64 + ...
B(x) = 1 + x/(1-x) + x^2/(1-2*x)^2 + x^3/(1-3*x)^3 + x^4/(1-4*x)^4 + x^5/(1-5*x)^5 + x^6/(1-6*x)^6 + x^7/(1-7*x)^7 + x^8/(1-8*x)^8 + ...
where
B(x) = 1 + x + 2*x^2 + 6*x^3 + 23*x^4 + 104*x^5 + 537*x^6 + 3100*x^7 + 19693*x^8 + 136064*x^9 + 1013345*x^10 + 8076644*x^11 + ... + A080108(n)*x^n + ...

Crossrefs

Cf. A080108.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff( sum(m=0, #A, x^m*( 1/(1-m*x +x*O(x^#A))^m - Ser(A)^(m^2)) ), #A) ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
Previous Showing 31-32 of 32 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.