A084081 -id:A084081 - cp's OEIS frontend

A093951 Sum of integers generated by n-1 substitutions, starting with 1, k -> k+1, k-1, .., 1.

1, 2, 4, 8, 17, 36, 80, 176, 403, 910, 2128, 4896, 11628, 27132, 65208, 153824, 373175, 888030, 2170740, 5202600, 12797265, 30853680, 76292736, 184863168, 459162452, 1117370696, 2786017120, 6804995008, 17024247304, 41717833740, 104673837384

Offset: 1

Wouter Meeussen, Apr 18 2004

nonn

Substitutions 1 -> {2}, 2 -> {3,1}, 3 -> {4,2}, 4 -> {5,3,1}, 5 -> {6,4,2}, 6 -> {7,5,3,1}, 7 -> {8,6,4,2}, etc. The function f(n) gives determinant of (I_{n} - x * A(n)) where I_{n} is the identity matrix and A(n) = 0 if j > i + 1 otherwise (i+j) mod 2, for i = 1, ..., n and j = 1, ..., n, and can be written in terms of Dickson polynomials as g(w) = x*D_(w-1)(1+x, x*(1+x)) + (1-2*x)*E_(w-1)(1+x, x*(1+x)). - Francisco Salinas (franciscodesalinas(AT)hotmail.com), Apr 13 2004
Count of integers is A047749.
Sum of integers with substitution starting from 0 is A084081.

			GF(12) = (1 + 2*x - 7*x^2 - 14*x^3 + 9*x^4 + 20*x^5 + 2*x^6 - 2*x^7 + 2*x^11)/(1 - 11*x^2 + 36*x^4 - 35*x^6 + 5*x^8) produces a(1) to a(12).
a(4)=8 since 4-1 = 3 substitutions on 1 produce 1 -> 2 -> 3+1 -> 4 + 2 + 2 = 8.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A006629, A047749, A056096, A084081.

function A093951(n)
  if (n mod 2) eq 0 then return 8*Binomial(Floor(3*n/2), Floor((n-2)/2))/(n+2);
  else return 6*Binomial(Floor((3*n+1)/2), Floor((n-1)/2))/(n+2) - 2*Binomial(Floor((3*n-1)/2), Floor((n-1)/2))/(n+1);
  end if; return A093951;
end function;
[A093951(n): n in [1..40]]; // G. C. Greubel, Oct 17 2022

Plus@@@Flatten/@NestList[ #/.k_Integer:>Range[k+1, 1, -2]&, {1}, 8];(*or for n>16 *); f[1]=1; f[2]=1-x^2; f[3]=1-2x^2; f[n_]:=f[n]=Expand[f[n-1]-x^2 f[n-3]]; g[1]=1; g[2]=1+2x; g[3]=1+2x+2x^2; g[n_]:=g[n]=Expand[g[n-1] -x^2 g[n-3]+2 x^(n-1)]; GF[n_]:=g[n]/f[n]; CoefficientList[Series[GF[36], {x, 0, 36}], x]

{a(n)=if(n%2==0,4*binomial(3*n/2,n/2-1)/(n/2+1), 6*binomial(3*(n\2)+2, n\2)/(2*(n\2)+3) - binomial(3*(n\2)+1,n\2)/(n\2+1))} \\ Paul D. Hanna, Apr 24 2006

def A093951(n):
    if (n%2==0): return 8*binomial(3*n/2, (n-2)/2)/(n+2)
    else: return 6*binomial((3*n+1)/2, (n-1)/2)/(n+2) - 2*binomial((3*n-1)/2, (n-1)/2)/(n+1)
[A093951(n) for n in range(1,40)] # G. C. Greubel, Oct 17 2022

a(n) = [x^n] GF(n) with GF(n) = g(n)/f(n) and f(1)=1, f(2)=1-x^2, f(3)=1-2*x^2, f(n) = f(n-1) - x^2*f(n-3) and g(1)=1, g(2)=1+2*x, g(3)=1+2*x+2*x^2, g(n) = g(n-1) - x^2*g(n-3) + 2*x^(n-1).
From Paul D. Hanna, Apr 24 2006: (Start)
a(2*n) = 4*binomial(3*n, n-1)/(n+1) = 2*A006629(n-1).
a(2*n+1) = 6*binomial(3*n+2, n)/(2*n+3) - binomial(3*n+1, n)/(n+1) = A056096(n+3). (End)

A124816 Product of Riordan array (1,x(1-x^2))^(-1) and number triangle T(n,k)=C(floor(k/2),n-k).

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 2, 1, 1, 0, 3, 3, 3, 2, 1, 0, 0, 7, 4, 5, 2, 1, 0, 12, 12, 12, 10, 6, 3, 1, 0, 0, 30, 18, 24, 12, 9, 3, 1, 0, 55, 55, 55, 50, 32, 22, 10, 4, 1, 0, 0, 143, 88, 121, 66, 57, 25, 14, 4, 1, 0, 273, 273, 273

Offset: 0

Paul Barry, Nov 08 2006

Product of A124305 and A124788. Columns include aerated A001764,A047749(n+1),A124817,A084081. Row sums are A124818.

			Triangle begins
1,
0, 1,
0, 0, 1,
0, 1, 1, 1,
0, 0, 2, 1, 1,
0, 3, 3, 3, 2, 1,
0, 0, 7, 4, 5, 2, 1,
0, 12, 12, 12, 10, 6, 3, 1,
0, 0, 30, 18, 24, 12, 9, 3, 1,
0, 55, 55, 55, 50, 32, 22, 10, 4, 1

Cf. A124790.

cp's OEIS Frontend

A093951 Sum of integers generated by n-1 substitutions, starting with 1, k -> k+1, k-1, .., 1.

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