A139375
A Fibonacci-Catalan triangle. Also called the Fibonacci triangle.
Original entry on oeis.org
1, 1, 1, 2, 2, 1, 3, 5, 3, 1, 5, 12, 9, 4, 1, 8, 31, 26, 14, 5, 1, 13, 85, 77, 46, 20, 6, 1, 21, 248, 235, 150, 73, 27, 7, 1, 34, 762, 741, 493, 258, 108, 35, 8, 1, 55, 2440, 2406, 1644, 903, 410, 152, 44, 9, 1, 89, 8064, 8009
Offset: 0
Triangle begins
1,
1, 1,
2, 2, 1,
3, 5, 3, 1,
5, 12, 9, 4, 1,
8, 31, 26, 14, 5, 1,
13, 85, 77, 46, 20, 6, 1,
21, 248, 235, 150, 73, 27, 7, 1,
34, 762, 741, 493, 258, 108, 35, 8, 1
The production matrix for this array is
1, 1,
1, 1, 1,
-1, 1, 1, 1,
0, 1, 1, 1, 1,
0, 1, 1, 1, 1, 1,
0, 1, 1, 1, 1, 1, 1,
0, 1, 1, 1, 1, 1, 1
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RIORDAN := proc(d,h,n,k)
d*h^k ;
expand(%) ;
coeftayl(%,x=0,n) ;
end proc:
A139375 := proc(n,k)
RIORDAN(1/(1-x-x^2),(1-sqrt(1-4*x))/2,n,k) ;
end proc: # R. J. Mathar, Jul 09 2013
-
T[n_, 0]:= Fibonacci[n + 1]; T[n_, k_]:= k*Sum[Fibonacci[i + 1]*Binomial[2*(n - i) - k - 1, n - i - 1]/(n - i), {i, 0, n - k}]; Table[T[n, k], {n, 0, 49}, {k, 0, n}] // Flatten (* G. C. Greubel, Oct 20 2016 *)
A089867
Permutation of natural numbers induced by the Catalan bijection gma089867 acting on the parenthesizations/binary trees encoded by A014486/A063171.
Original entry on oeis.org
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 21, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 49, 51, 52, 53, 54, 55, 60, 61, 64, 63, 56, 57, 59, 58, 62, 65, 66, 67, 68, 69
Offset: 0
Number of cycles:
A089846. Number of fixed-points:
A090826. Max. cycle size:
A086586. LCM of cycle sizes:
A086587. (In range [
A014137(n-1)..
A014138(n-1)] of this permutation, possibly shifted one term left or right).
A089868
Permutation of natural numbers induced by the Catalan bijection gma089868 acting on the parenthesizations/binary trees encoded by A014486/A063171.
Original entry on oeis.org
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 21, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 49, 51, 52, 53, 54, 55, 60, 61, 63, 62, 56, 57, 64, 59, 58, 65, 66, 67, 68, 69
Offset: 0
Number of cycles:
A089846. Number of fixed-points:
A090826. Max. cycle size:
A086586. LCM of cycle sizes:
A086587. (In range [
A014137(n-1)..
A014138(n-1)] of this permutation, possibly shifted one term left or right).
A277220
Exponential convolution of Fibonacci (A000045) and Catalan (A000108) numbers.
Original entry on oeis.org
0, 1, 3, 11, 43, 180, 790, 3590, 16745, 79705, 385615, 1890747, 9375216, 46931897, 236873261, 1204089630, 6159064015, 31678706490, 163739008070, 850051218980, 4430529313065, 23175017046351, 121617754070653, 640122809255716, 3378402106118508, 17875011275340275
Offset: 0
-
[(&+[Binomial(n,k)*Fibonacci(k)*Catalan(n-k): k in [0..n]]): n in [0..30]]; // G. C. Greubel, Oct 22 2018
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Table[Sum[Binomial[n, k] Fibonacci[k] CatalanNumber[n - k], {k, 0, n}], {n, 0, 30}] (* or *)
Round@Table[(GoldenRatio^n Hypergeometric2F1[1/2, -n, 2, -4/GoldenRatio] - (-GoldenRatio)^(-n) Hypergeometric2F1[1/2, -n, 2, 4 GoldenRatio])/Sqrt[5], {n, 0, 30}] (* Round is equivalent to FullSimplify here, but is much faster *)
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for(n=0, 30, print1(sum(k=0,n, binomial(n,k)*fibonacci(k)* binomial(2*n-2*k,n-k)/(n-k+1)), ", ")) \\ G. C. Greubel, Oct 22 2018
A277251
Exponential convolution of Lucas (A000032) and Catalan (A000108) numbers.
Original entry on oeis.org
2, 3, 9, 29, 107, 430, 1840, 8230, 38015, 179873, 867079, 4242111, 21006358, 105072063, 530058079, 2693632580, 13775807415, 70847283680, 366167521240, 1900884870494, 9907318315587, 51822028122623, 271949090063769, 1431369293422604, 7554372307564282
Offset: 0
-
Table[Sum[Binomial[n, k] LucasL[k] CatalanNumber[n - k], {k, 0, n}], {n, 0,
30}] (* or *)
Round@Table[GoldenRatio^n Hypergeometric2F1[1/2, -n, 2, -4/GoldenRatio] + (-GoldenRatio)^(-n) Hypergeometric2F1[1/2, -n, 2, 4 GoldenRatio], {n, 0, 30}] (* Round is equivalent to FullSimplify here, but is much faster *)
Showing 1-5 of 5 results.
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