A006012 -id:A006012 - cp's OEIS frontend

A367564 Triangular array read by rows: T(n, k) = binomial(n, k) * A001333(n - k).

1, 1, 1, 3, 2, 1, 7, 9, 3, 1, 17, 28, 18, 4, 1, 41, 85, 70, 30, 5, 1, 99, 246, 255, 140, 45, 6, 1, 239, 693, 861, 595, 245, 63, 7, 1, 577, 1912, 2772, 2296, 1190, 392, 84, 8, 1, 1393, 5193, 8604, 8316, 5166, 2142, 588, 108, 9, 1, 3363, 13930, 25965, 28680, 20790, 10332, 3570, 840, 135, 10, 1

Offset: 0

Peter Luschny, Nov 25 2023

			Triangle T(n, k) starts:
[0]    1;
[1]    1,    1;
[2]    3,    2,    1;
[3]    7,    9,    3,    1;
[4]   17,   28,   18,    4,    1;
[5]   41,   85,   70,   30,    5,    1;
[6]   99,  246,  255,  140,   45,    6,   1;
[7]  239,  693,  861,  595,  245,   63,   7,   1;
[8]  577, 1912, 2772, 2296, 1190,  392,  84,   8, 1;
[9] 1393, 5193, 8604, 8316, 5166, 2142, 588, 108, 9, 1;

Cf. A001333 (column 0), A006012 (row sums), A367211.

P := proc(n) option remember; ifelse(n <= 1, 1, 2*P(n - 1) + P(n - 2)) end:
T := (n, k) -> P(n - k) * binomial(n, k):
for n from 0 to 9 do seq(T(n, k), k = 0..n) od;

P[n_] := P[n] = If[n <= 1, 1, 2 P[n - 1] + P[n - 2]];
T[n_, k_] := P[n - k] Binomial[n, k];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Oct 10 2024, after Peter Luschny *)

From Werner Schulte, Nov 26 2023: (Start)
The row polynomials p(n, x) = Sum_{k=0..n} T(n, k) * x^k satisfy:
  a) p'(n, x) = n * p(n-1, x) where p' is the first derivative of p;
  b) p(0, x) = 1, p(1, x) = 1 + x and p(n, x) = (2+2*x) * p(n-1, x) + (1-2*x-x^2) * p(n-2, x) for n > 1.
T(n, 0) = A001333(n) for n >= 0 and T(n, k) = T(n-1, k-1) * n / k for 0 < k <= n.
G.f.: (1 - (1+x) * t) / (1 - (2+2*x) * t - (1-2*x-x^2) * t^2). (End)

A377501 a(n) = 2 + 4^(n - 1) - (2 - sqrt(2))^(n - 1) - (2 + sqrt(2))^(n - 1).

Original entry on oeis.org

1, 2, 6, 26, 122, 562, 2514, 10978, 47074, 199106, 833346, 3459458, 14268290, 58542850, 239189250, 973889026, 3954048514, 16015899650, 64745436162, 261309683714, 1053186816002, 4239883710466, 17052184465410, 68525063462914, 275180257009666, 1104408389468162

Offset: 1

Eric W. Weisstein, Oct 30 2024

a(n) is also the number of edge cuts in the wheel graph on n vertices for n > 3.

Eric Weisstein's World of Mathematics, Edge Cut.
Eric Weisstein's World of Mathematics, Wheel Graph.
Index entries for linear recurrences with constant coefficients, signature (9,-26,26,-8).

Cf. A158525.

Table[2 + 4^(n - 1) - (2 - Sqrt[2])^(n - 1) - (2 + Sqrt[2])^(n - 1), {n, 26}]
LinearRecurrence[{9, -26, 26, -8}, {1, 2, 6, 26}, 20]
CoefficientList[Series[-(-1 + 7 x - 14 x^2 + 2 x^3)/((-1 + x) (-1 + 4 x) (1 - 4 x + 2 x^2)), {x, 0, 20}], x]

a(n) = 2 + 4^(n - 1) - (2 - sqrt(2))^(n - 1) - (2 + sqrt(2))^(n - 1) = 2+4^(n-1)-2*A006012(n-1).
a(n) = 9*a(n-1)-26*a(n-2)+26*a(n-3)-8*a(n-4).
G.f.: -x*(-1+7*x-14*x^2+2*x^3)/((-1+x)*(-1+4*x)*(1-4*x+2*x^2)).
a(n) = 2^(2*(n-1))-A158525(n) for n >= 4. - Pontus von Brömssen, Nov 06 2024
E.g.f.: exp(2*x)*(-2*cosh(sqrt(2)*x) - 2*sinh(x) + cosh(x)*(2 + sinh(x)) + sqrt(2)*sinh(sqrt(2)*x)). - Stefano Spezia, Nov 08 2024

cp's OEIS Frontend

A367564 Triangular array read by rows: T(n, k) = binomial(n, k) * A001333(n - k).

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