A290707 -id:A290707 - cp's OEIS frontend

A081494 Start with Pascal's triangle; form a triangle by sliding down n steps from top on both sides and including the horizontal row, deleting the inner numbers; a(n) = sum of entries on perimeter of triangle.

1, 3, 7, 13, 23, 41, 75, 141, 271, 529, 1043, 2069, 4119, 8217, 16411, 32797, 65567, 131105, 262179, 524325, 1048615, 2097193, 4194347, 8388653, 16777263, 33554481, 67108915, 134217781, 268435511, 536870969, 1073741883, 2147483709

Offset: 1

Amarnath Murthy, Mar 25 2003

nonn

			The triangle pertaining to n = 4 is obtained from the solid triangle
        1
      1   1
    1   2   1
  1   3   3   1
giving
        1
      1   1
    1       1
  1   3   3   1
and the sum of all the numbers is 13, so a(4) = 13.

Cf. A081495, A081496, A081497.

First differences of A290707.

restart:a:= proc(n) option remember; if n=0 then 1 else add((binomial (n,j)+2), j=0..n-1) fi end: seq (a(n), n=0..31); # Zerinvary Lajos, Mar 29 2009

For n > 1, a(n) = A061761(n-1). - David Wasserman, Jun 03 2004

Corrected and extended by David Wasserman, Jun 03 2004

A290818 Array read by antidiagonals: T(m,n) = number of irredundant sets in the lattice (rook) graph K_m X K_n.

Original entry on oeis.org

2, 3, 3, 4, 11, 4, 5, 24, 24, 5, 6, 47, 94, 47, 6, 7, 88, 272, 272, 88, 7, 8, 163, 774, 1185, 774, 163, 8, 9, 304, 2230, 4280, 4280, 2230, 304, 9, 10, 575, 6542, 15781, 20106, 15781, 6542, 575, 10, 11, 1104, 19452, 60604, 88512, 88512, 60604, 19452, 1104, 11

Offset: 1

Andrew Howroyd, Aug 11 2017

			Array begins:
===============================================================
m\n| 1   2     3      4       5        6        7         8
---+-----------------------------------------------------------
1  | 2   3     4      5       6        7        8         9 ...
2  | 3  11    24     47      88      163      304       575 ...
3  | 4  24    94    272     774     2230     6542     19452 ...
4  | 5  47   272   1185    4280    15781    60604    240073 ...
5  | 6  88   774   4280   20106    88512   400728   1879744 ...
6  | 7 163  2230  15781   88512   453271  2326534  12363513 ...
7  | 8 304  6542  60604  400728  2326534 13169346  76446456 ...
8  | 9 575 19452 240073 1879744 12363513 76446456 476777153 ...
...

Andrew Howroyd, Table of n, a(n) for n = 1..1275
Eric Weisstein's World of Mathematics, Irredundant Set
Eric Weisstein's World of Mathematics, Rook Graph

Row 2 is A290707 for n > 1.
Main diagonal is A290586.
Cf. A287274, A290632.

s[n_, k_]:=Sum[(-1)^i*Binomial[n, i] StirlingS2[n - i, k - i], {i, 0, Min[n, k]}];
c[m_, n_, x_]:=Sum[Binomial[m, i] (n^i - n!*StirlingS2[i, n])*x^i, {i, 0, m - 1}];
p[m_, n_, x_]:=Sum[Sum[Binomial[m, k] Binomial[n, r]* k!*s[r, k]*x^r*c[m - k, n - r, x], {r, 2k, n - 1}], {k,0, m - 1}];
b[m_, n_, x_]:=m^n*x^n + n^m*x^m - If[n<=m, n!*x^m*StirlingS2[m, n], m!*x^n*StirlingS2[n, m]];
T[m_, n_]:= b[m, n, 1] + p[m, n, 1];
Table[T[n, m -n + 1], {m, 10}, {n, m}]//Flatten
(* Indranil Ghosh, Aug 12 2017, after PARI code *)

\\ See A. Howroyd note in A290586 for explanation.
s(n,k)=sum(i=0, min(n, k), (-1)^i * binomial(n, i) * stirling(n-i, k-i, 2) );
c(m,n,x)=sum(i=0, m-1, binomial(m, i) * (n^i - n!*stirling(i, n, 2))*x^i);
p(m,n,x)={sum(k=0, m-1, sum(r=2*k, n-1, binomial(m, k) * binomial(n, r) * k! * s(r, k) * x^r * c(m-k, n-r, x) ))}
b(m,n,x) = m^n*x^n + n^m*x^m - if(n<=m, n!*x^m*stirling(m, n, 2), m!*x^n*stirling(n, m, 2));
T(m,n) = b(m,n,1) + p(m,n,1);
for(m=1,10,for(n=1,m,print1(T(n,m-n+1),", ")));

T(m,n) = A290632(m, n) + Sum_{k=0..m-1} Sum_{r=2*k..n-1} binomial(m,k) * binomial(n,r) * k! * A008299(r,k) * c(m-k,n-r) where c(m,n) = Sum_{i=0..m-1} binomial(n,i) * (n^i - n!*stirling2(i, n)).

A290709 Number of irredundant sets in the complete tripartite graph K_{n,n,n}.

Original entry on oeis.org

4, 22, 49, 94, 169, 298, 529, 958, 1777, 3370, 6505, 12718, 25081, 49738, 98977, 197374, 394081, 787402, 1573945, 3146926, 6292777, 12584362, 25167409, 50333374, 100665169, 201328618, 402655369, 805308718, 1610615257, 3221228170, 6442453825, 12884904958

Offset: 1

Eric W. Weisstein, Aug 09 2017

nonn

When n > 1, the nonempty irredundant sets are those consisting of either any number of vertices from a single partition or otherwise exactly two vertices from different partitions. - Andrew Howroyd, Aug 10 2017

Eric Weisstein's World of Mathematics, Complete Tripartite Graph
Eric Weisstein's World of Mathematics, Irredundant Set
Index entries for linear recurrences with constant coefficients, signature (5, -9, 7, -2).

Cf. A290707.

Table[If[n == 1, 4, 3 (2^n + n^2) - 2], {n, 20}]
Join[{4}, LinearRecurrence[{5, -9, 7, -2}, {22, 49, 94, 169}, 20]]
CoefficientList[Series[(4 + 2 x - 25 x^2 + 19 x^3 - 6 x^4)/((-1 + x)^3 (-1 + 2 x)), {x, 0, 20}], x]

a(n) = if(n==1, 4, 3*(2^n + n^2) - 2); \\ Andrew Howroyd, Aug 10 2017

a(n) = 3*(2^n + n^2) - 2 for n > 1. - Andrew Howroyd, Aug 10 2017
a(n) = 5*a(n-1) - 9*a(n-2) + 7*a(n-3) - 2*a(n-4) for n > 5.
G.f.: (x (4 + 2 x - 25 x^2 + 19 x^3 - 6 x^4))/((-1 + x)^3 (-1 + 2 x)).

a(7)-a(32) from Andrew Howroyd, Aug 10 2017

cp's OEIS Frontend

A081494 Start with Pascal's triangle; form a triangle by sliding down n steps from top on both sides and including the horizontal row, deleting the inner numbers; a(n) = sum of entries on perimeter of triangle.

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A290818 Array read by antidiagonals: T(m,n) = number of irredundant sets in the lattice (rook) graph K_m X K_n.

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A290709 Number of irredundant sets in the complete tripartite graph K_{n,n,n}.

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Mathematica

PARI

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Extensions