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A325983 Row sums of the triangle A325982. 3
1, 1, 2, 2, 5, 5, 18, 21, 77, 102, 337, 480, 1449, 2155, 6107, 9348, 25355, 39639, 104188, 165596, 425156, 684926, 1726737, 2813582, 6990175, 11501905, 28232753, 46854161, 113841632, 190362483, 458480128, 771855377, 1844765161, 3124639626, 7417428613, 12633074088 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

LINKS

Stefano Spezia, Table of n, a(n) for n = 1..2000

FORMULA

a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n - 1, k - 1) - binomial(n - k - 1, k - 1) + 1.

a(n) = Sum_{k=0..A004526(n-1)} A007318(n - 1, k - 1) - A007318(n - k - 1, k - 1) + 1.

Conjecture: a(n) ~ 2^(n-2). - Stefano Spezia, Jun 16 2021

This conjecture is true. Its proof follows from Vaclav Kotesovec's formula used in the 2nd Mathematica code. - Stefano Spezia, Jun 27 2021

Recurrence: n*(15*n^6 - 600*n^5 + 9874*n^4 - 85078*n^3 + 403791*n^2 - 1000762*n + 1013720)*a(n) = (45*n^7 - 1815*n^6 + 30507*n^5 - 273677*n^4 + 1393992*n^3 - 3930412*n^2 + 5362320*n - 2240000)*a(n-1) + 2*(30*n^7 - 1230*n^6 + 20813*n^5 - 186314*n^4 + 944533*n^3 - 2692476*n^2 + 3985604*n - 2395680)*a(n-2) - (255*n^7 - 10470*n^6 + 180608*n^5 - 1685204*n^4 + 9136801*n^3 - 28646510*n^2 + 47833000*n - 32569600)*a(n-3) + (15*n^7 - 585*n^6 + 11089*n^5 - 129179*n^4 + 931040*n^3 - 3910812*n^2 + 8540192*n - 7271040)*a(n-4) + 2*(165*n^7 - 6810*n^6 + 118529*n^5 - 1121692*n^4 + 6211866*n^3 - 20094734*n^2 + 35150196*n - 25693920)*a(n-5) - 4*(15*n^7 - 600*n^6 + 10309*n^5 - 98168*n^4 + 555848*n^3 - 1857428*n^2 + 3364504*n - 2524480)*a(n-6) - 8*(n-7)*(15*n^6 - 510*n^5 + 7099*n^4 - 51282*n^3 + 202026*n^2 - 411828*n + 340960)*a(n-7). - Vaclav Kotesovec, Jun 20 2021

MAPLE

a := n -> add(binomial(n-1, k-1)-binomial(n-k-1, k-1)+1, k = 0 .. floor((n-1)/2)): seq(a(n), n = 1 .. 40);

MATHEMATICA

T[n_, k_]:=Binomial[n-1, k-1]-Binomial[n-k-1, k-1]+1; a[n_]:=Sum[T[n, k], {k, 0, Floor[(n-1)/2]}]; Array[a, 40]

Table[If[EvenQ[n], 2^(n - 2) + n/2 + 1 - Binomial[n, n/2]/2 + Fibonacci[n]/2 - LucasL[n]/2, 2^(n - 2) + (n + 1)/2 - Binomial[n - 1, (n - 1)/2]/2 - Fibonacci[n - 3] - 3*Fibonacci[n]/2 + LucasL[n]/2], {n, 1, 40}] (* Vaclav Kotesovec, Jun 20 2021 *)

PROG

(GAP) List([1..40], n->Sum([0..Int((n-1)/2)], k->Binomial(n-1, k-1)-Binomial(n-k-1, k-1)+1));

(MAGMA) [(&+[Binomial(n-1, k-1)-Binomial(n-k-1, k-1)+1: k in [0..Floor((n-1)/2)]]): n in [1..40]];

(PARI) a(n) = sum(k=0, floor((n-1)/2), binomial(n - 1, k - 1) - binomial(n - k - 1, k - 1) + 1);

CROSSREFS

Cf. A004526, A007318, A325982.

Sequence in context: A245844 A083849 A326512 * A063501 A103892 A000403

Adjacent sequences:  A325980 A325981 A325982 * A325984 A325985 A325986

KEYWORD

nonn

AUTHOR

Stefano Spezia, May 29 2019

STATUS

approved

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Last modified September 21 16:31 EDT 2021. Contains 347598 sequences. (Running on oeis4.)