A053199 -id:A053199 - cp's OEIS frontend

A053297 Row sums of array T in A053199.

1, 5, 22, 92, 372, 1468, 5688, 21728, 82064, 307088, 1140320, 4206912, 15434048, 56350912, 204875648, 742104064, 2679197952, 9644109056, 34623075840, 124001176576, 443136848896, 1580464036864, 5626501838848, 19996918849536, 70960191213568, 251445325991936

Offset: 1

Clark Kimberling, Mar 18 2000

nonn

The generating series is a power series composition G(F(t)) where F(t) = t + 3*t^2 + 7*t^3 + 15*t^4 + ... is generating series of A000225, and G(t) = t + 2*t^2 + 3*t^3 + 4*t^4 + ... is generating series of the natural numbers A000027. Proof follows as in reference below. - Oboifeng Dira, Nov 03 2016

			G.f. = x + 5*x^2 + 22*x^3 + 92*x^4 + 372*x^5 + 1468*x^6 + 5688*x^7 + 21728*x^8 + ...

G. C. Greubel, Table of n, a(n) for n = 1..1000
O. Dira, A Note on Composition and Recursion, Southeast Asian Bulletin of Mathematics . 2017, Vol. 41 Issue 6, pp. 849-853.

Cf. A000027, A000225, A053199.

m:=25; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(x*(1-x)*(1-2*x)/(1-4*x+2*x^2)^2)); // G. C. Greubel, May 24 2018

Drop[CoefficientList[Series[x*(1-x)*(1-2*x)/(1-4*x+2*x^2)^2, {x,0,50}], x], 1] (* G. C. Greubel, May 24 2018 *)

my(x='x+O('x^30)); Vec(x*(1-x)*(1-2*x)/(1-4*x+2*x^2)^2) \\ G. C. Greubel, May 24 2018

G.f.: x * (1 - x) * (1 - 2*x) / (1 - 4*x + 2*x^2)^2. - Michael Somos, Nov 03 2016
a(n) = 8*a(n-1) + 20*a(n-2) - 16*a(n-3) + 4*a(n-4) for all n in Z. - Michael Somos, Nov 03 2016
a(n) = -a(-n) * 2^n for all n in Z. - Michael Somos, Nov 03 2016

A054144 Triangular array T: put T(n,0)=n for all n >= 0 and all other T(n,k)=0; then put T(n,k)=Sum{T(i,j): 0<=j<=i-n+k, n-k<=i<=n}.

Original entry on oeis.org

0, 1, 1, 2, 3, 7, 3, 5, 14, 36, 4, 7, 21, 60, 164, 5, 9, 28, 84, 246, 700, 6, 11, 35, 108, 328, 980, 2868, 7, 13, 42, 132, 410, 1260, 3824, 11424, 8, 15, 49, 156, 492, 1540, 4780, 14688, 44576, 9, 17, 56, 180, 574, 1820, 5736, 17952, 55720, 171216

Offset: 0

Clark Kimberling, Mar 18 2000

The main diagonal is A181292. The triangle below the main diagonal is A053199 which starts with T(n,0) = n + 1 instead of T(n,0) = n here. - Georg Fischer, Nov 16 2021

			Rows: {0}, {1,1}, {2,3,7}, {3,5,14,36}, ...

Cf. A053199, A181292.

A335436 Triangle read by rows: T(n,k) = 2*n+1 for k = 0 and otherwise T(n,k) = Sum_{i=n-k..n, j=0..i-n+k, i<>n or j<>k} T(i,j).

Original entry on oeis.org

1, 3, 4, 5, 8, 21, 7, 12, 35, 96, 9, 16, 49, 144, 410, 11, 20, 63, 192, 574, 1680, 13, 24, 77, 240, 738, 2240, 6692, 15, 28, 91, 288, 902, 2800, 8604, 26112, 17, 32, 105, 336, 1066, 3360, 10516, 32640, 100296, 19, 36, 119, 384, 1230, 3920, 12428, 39168, 122584, 380480

Offset: 0

Oboifeng Dira, Jul 14 2020

			Triangle begins:
  1;
  3,  4;
  5,  8, 21;
  7, 12, 35,  96;
  9, 16, 49, 144, 410;
  ...
T(3,2) = ((2+sqrt(2))^3-(2-sqrt(2))^3)*(6-2+1)/(4*sqrt(2)) = (28*sqrt(2))*(5)/(4*sqrt(2)) = 35.

Oboifeng Dira, Table of n, a(n) for n = 0..54

Cf. A053199, A294285, A064339.

T := proc (n, k) if k = 0 and 0 <= n then 2*n+1 elif 1 <= k and k <= n then round((((2+sqrt(2))^(k+1)-(2-sqrt(2))^(k+1))*(2*n-k+1)/(4*sqrt(2)))) else 0 end if end proc:seq(print(seq(T(n, k), k=0..n)), n=0..9);

T(n,k) = if (k==0, 2*n+1, if (k<=n, sum(i=n-k, n, sum(j=0, i-n+k, if ((i==n) && (j==k), 0, T(i,j)), 0))));
matrix(10, 10, n, k, T(n-1, k-1)) \\ Michel Marcus, Sep 08 2020

T(n, k) = if (k==0, 2*n+1, if (k>n, 0, my(w=quadgen(8, 'w)); ((2+w)^(k+1)-(2-w)^(k+1))*(2*n-k+1)/(4*w)));
matrix(10, 10, n, k, T(n-1, k-1)) \\ Michel Marcus, Sep 10 2020

T(n,0) = 2*n+1 for k=0;

T(n,k) = ((2+sqrt(2))^(k+1)-(2-sqrt(2))^(k+1))*(2*n-k+1)/(4*sqrt(2)) for 1<=k<=n.

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A053297 Row sums of array T in A053199.

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A054144 Triangular array T: put T(n,0)=n for all n >= 0 and all other T(n,k)=0; then put T(n,k)=Sum{T(i,j): 0<=j<=i-n+k, n-k<=i<=n}.

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A335436 Triangle read by rows: T(n,k) = 2*n+1 for k = 0 and otherwise T(n,k) = Sum_{i=n-k..n, j=0..i-n+k, i<>n or j<>k} T(i,j).

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