A176332 -id:A176332 - cp's OEIS frontend

A176335 Central coefficients T(2n,n) of number triangle A176331.

1, 3, 28, 315, 3876, 50358, 678112, 9365499, 131809060, 1882294128, 27193657008, 396600597198, 5829739893264, 86262567856650, 1283677784658528, 19196304797150715, 288295493121264420, 4346056823245242420

Offset: 0

Paul Barry, Apr 15 2010

G. C. Greubel, Table of n, a(n) for n = 0..825
F. Baldassarri, S. Bosch, B. Dwork, (eds), p-adic Analysis. Lecture Notes in Mathematics, vol. 1454, pp. 194 - 204, Springer, Berlin, Heidelberg.
Matthijs J. Coster, Supercongruences.

Cf. A112029, A176331, A176332, A176334.

T:= function(n,k)
    return Sum([0..n], j-> (-1)^(n-j)*Binomial(j,k)*Binomial(j,n-k) );
  end;
List([0..30], n-> T(2*n,n) ); # G. C. Greubel, Dec 07 2019

T:= func< n,k | &+[(-1)^(n-j)*Binomial(j,n-k)*Binomial(j,k): j in [0..n]] >;
[T(2*n,n): n in [0..30]]; // G. C. Greubel, Dec 07 2019

A176335 := proc(n)
    add((-1)^k*binomial(k,n)^2,k=0..2*n);
end proc: # R. J. Mathar, Feb 10 2015

T[n_, k_]:= Sum[(-1)^(n-j)*Binomial[j, k]*Binomial[j, n-k], {j,0,n}]; Table[T[2*n, n], {n,0,30}] (* G. C. Greubel, Dec 07 2019 *)

T(n,k) = sum(j=0, n, (-1)^(n-j)*binomial(j, n-k)*binomial(j, k));
vector(31, n, T(2*(n-1), n-1) ) \\ G. C. Greubel, Dec 07 2019

@CachedFunction
def T(n, k): return sum( (-1)^(n-j)*binomial(j, n-k)*binomial(j, k) for j in (0..n))
[T(2*n, n) for n in (0..30)] # G. C. Greubel, Dec 07 2019

a(n) = Sum_{k=0..2n} C(k,n)^2*(-1)^k.
Conjecture: 224*n^2*(n-1)*a(n) - 48*(n-1)*(65*n^2 - 36*n - 13)*a(n-1) + 4*(-1839*n^3 + 11081*n^2 - 21932*n + 14280)*a(n-2) + 12*(-81*n^3 + 326*n^2 - 591*n + 562)*a(n-3) - (n-3)*(1853*n^2 - 7403*n + 7140)*a(n-4) - 12*(n-4)*(2*n-7)^2*a(n-5) = 0. - R. J. Mathar, Feb 10 2015
From Peter Bala, Aug 08 2024: (Start)
a(n) = Sum_{k = 0..n} (-1)^(n+k)*binomial(n+k, k)^2. Cf. A112029.
Conjecture (assuming an offset of 1): the supercongruence a(n*p^r) == a(n*p^(r-1)) (mod p^(3*r)) holds for all primes p >= 5 and all positive integers n and r [added Nov 29 2024: proved by Coster. See Theorem 4]. (End)
a(n) ~ 2^(4*n+2) / (5*Pi*n). - Vaclav Kotesovec, Aug 08 2024
a(n) = binomial(2*n, n)^2 * hypergeom([1, -n, -n], [-2*n, -2*n], -1). - Peter Luschny, Nov 29 2024

A176331 Triangle read by rows: T(n, k) = Sum_{j=0..n} C(j, n-k) * C(j, k) * (-1)^(n - j).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 7, 7, 1, 1, 13, 28, 13, 1, 1, 21, 79, 79, 21, 1, 1, 31, 181, 315, 181, 31, 1, 1, 43, 361, 971, 971, 361, 43, 1, 1, 57, 652, 2511, 3876, 2511, 652, 57, 1, 1, 73, 1093, 5713, 12606, 12606, 5713, 1093, 73, 1, 1, 91, 1729, 11789, 35246, 50358, 35246, 11789, 1729, 91, 1

Offset: 0

Paul Barry, Apr 15 2010

			Triangle begins
  1;
  1,  1;
  1,  3,    1;
  1,  7,    7,     1;
  1, 13,   28,    13,     1;
  1, 21,   79,    79,    21,     1;
  1, 31,  181,   315,   181,    31,     1;
  1, 43,  361,   971,   971,   361,    43,     1;
  1, 57,  652,  2511,  3876,  2511,   652,    57,    1;
  1, 73, 1093,  5713, 12606, 12606,  5713,  1093,   73,  1;
  1, 91, 1729, 11789, 35246, 50358, 35246, 11789, 1729, 91, 1;

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Row sums are A176332.
Diagonal sums are A176334.
Central coefficients T(2*n, n) are A176335.

T:= function(n,k)
    return Sum([0..n], j-> (-1)^(n-j)*Binomial(j,k)*Binomial(j,n-k) );
  end;
Flat(List([0..10], n-> List([0..n], k-> T(n,k) ))); # G. C. Greubel, Dec 07 2019

T:= func< n,k | &+[(-1)^(n-j)*Binomial(j,n-k)*Binomial(j,k): j in [0..n]] >;
[T(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Dec 07 2019

T:= proc(n, k) option remember; add( (-1)^(n-j)*binomial(j, n-k)*binomial(j, k), j=0..n); end: seq(seq(T(n, k), k=0..n), n=0..10); # G. C. Greubel, Dec 07 2019
T := (n, k) -> binomial(n, k)^2*hypergeom([1, -k, -n + k], [-n, -n], -1):
seq(seq(simplify(T(n, k)), k = 0..n), n = 0..9); # Peter Luschny, May 13 2024

T[n_, k_]:= Sum[(-1)^(n-j)*Binomial[j, k]*Binomial[j, n-k], {j,0,n}]; Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Dec 07 2019 *)

T(n,k) = sum(j=0, n, (-1)^(n-j)*binomial(j, n-k)*binomial(j, k)); \\ G. C. Greubel, Dec 07 2019

@CachedFunction
def T(n, k): return sum( (-1)^(n-j)*binomial(j, n-k)*binomial(j, k) for j in (0..n))
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Dec 07 2019

T(n, k) = T(n, n-k).

T(n, k) = binomial(n, k)^2*hypergeom([1, -k, -n + k], [-n, -n], -1). - Peter Luschny, May 13 2024

A176333 Expansion of (1-3x)/(1-4x+9*x^2).

Original entry on oeis.org

1, 1, -5, -29, -71, -23, 547, 2395, 4657, -2927, -53621, -188141, -269975, 613369, 4883251, 14012683, 12101473, -77708255, -419746277, -979610813, -140726759, 8253590281, 34280901955, 62841295291, -57162936431, -794223403343, -2662427185493, -3501698111885

Offset: 0

Paul Barry, Apr 15 2010

Hankel transform of A176332.

Michael De Vlieger, Table of n, a(n) for n = 0..2097
Beata Bajorska-Harapińska, Barbara Smoleń, Roman Wituła, On Quaternion Equivalents for Quasi-Fibonacci Numbers, Shortly Quaternaccis, Advances in Applied Clifford Algebras (2019) Vol. 29, 54.
Index entries for linear recurrences with constant coefficients, signature (4,-9).

Cf. A176332, A190958, A190967.

a:=[1,1];; for n in [3..30] do a[n]:=4*a[n-1]-9*a[n-2]; od; a; # G. C. Greubel, Dec 07 2019

I:=[1,1]; [n le 2 select I[n] else 4*Self(n-1) - 9*Self(n-2): n in [1..30]]; // G. C. Greubel, Dec 07 2019

seq(coeff(series((1-3*x)/(1-4*x+9*x^2), x, n+1), x, n), n = 0..30); # G. C. Greubel, Dec 07 2019

CoefficientList[Series[(1-3x)/(1-4x+9x^2),{x,0,30}],x] (* or *) LinearRecurrence[{4,-9},{1,1},30] (* Harvey P. Dale, Sep 17 2012 *)

my(x='x+O('x^30)); Vec((1-3*x)/(1-4*x+9*x^2)) \\ G. C. Greubel, Dec 07 2019

def A176333_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-3*x)/(1-4*x+9*x^2) ).list()
A176333_list(30) # G. C. Greubel, Dec 07 2019

a(n) = 3^n*( cos(2*n*atan(1/sqrt(5))) - sin(2*n*atan(1/sqrt(5)))/sqrt(5) ).
a(0)=1, a(1)=1, a(n) = 4*a(n-1) - 9*a(n-2). - Harvey P. Dale, Sep 17 2012
a(n) = -3*A190967(n) + A190967(n+1). - R. J. Mathar, May 04 2013

A176334 Diagonal sums of number triangle A176331.

Original entry on oeis.org

1, 1, 2, 4, 9, 21, 51, 124, 305, 755, 1879, 4698, 11792, 29694, 74984, 189811, 481498, 1223713, 3115200, 7942134, 20275362, 51823246, 132604193, 339644739, 870745187, 2234208932, 5737129623, 14742751524, 37909928908, 97543380598

Offset: 0

Paul Barry, Apr 15 2010

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A176331, A176332, A176335.

T:= function(n,k)
    return Sum([0..n], j-> (-1)^(n-j)*Binomial(j,k)*Binomial(j,n-k) );
  end;
List([0..30], n-> Sum([0..Int(n/2)], j-> T(n-j,j) )); # G. C. Greubel, Dec 07 2019

T:= func< n,k | &+[(-1)^(n-j)*Binomial(j,n-k)*Binomial(j,k): j in [0..n]] >;
[(&+[T(n-k,k): k in [0..Floor(n/2)]]): n in [0..30]]; // G. C. Greubel, Dec 07 2019

A176334 := proc(n)
    add(add(binomial(j,n-2*k)*binomial(j,k)*(-1)^(n-k-j),j=0..n-k), k=0..floor(n/2)) ;
end proc: # R. J. Mathar, Feb 10 2015

T[n_, k_]:= T[n, k]= Sum[(-1)^(n-j)*Binomial[j, k]*Binomial[j, n-k], {j,0,n}]; Table[Sum[T[n-k, k], {k,0,Floor[n/2]}], {n,0,30}] (* G. C. Greubel, Dec 07 2019 *)

T(n,k) = sum(j=0, n, (-1)^(n-j)*binomial(j, n-k)*binomial(j, k));
vector(30, n, sum(j=0, (n-1)\2, T(n-j-1,j)) ) \\ G. C. Greubel, Dec 07 2019

@CachedFunction
def T(n, k): return sum( (-1)^(n-j)*binomial(j, n-k)*binomial(j, k) for j in (0..n))
[sum(T(n-k, k) for k in (0..floor(n/2))) for n in (0..30)] # G. C. Greubel, Dec 07 2019

a(n) = Sum_{k=0..floor(n/2)} Sum_{j=0..n-k} C(j,n-2k)*C(j,k)*(-1)^(n-k-j).

a(n) ~ phi^(2*n+3) / (4*5^(1/4)*sqrt(Pi*n)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Aug 08 2024

A181933 a(n) = Sum_{k=0..n} binomial(n+k,k)sin(Pi(n+k)/2).

Original entry on oeis.org

0, 1, -3, 9, -30, 106, -385, 1421, -5304, 19966, -75658, 288222, -1102790, 4234868, -16312773, 63003869, -243896960, 946066678, -3676303578, 14308370014, -55768166380, 217640082188, -850345208538, 3325907590274, -13020993588680

Offset: 0

Robert G. Wilson v, Apr 02 2012

sign

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A026641, A176332.

f[n_] := Sum[ Binomial[n + k, k] Sin[Pi (n + k)/2], {k, 0, n}]; Array[f, 25, 0]

makelist(coeff(taylor(1/2*(sqrt(4*x+1)*(1+x)-3*x-1)/(sqrt(4*x+1)*(x^2+3*x+1)-4*x^2-5*x-1),x,0,20),x,n),n,0,20); /* Vladimir Kruchinin, Mar 28 2016 */

x='x+O('x^50); concat([0], Vec((1/2)*(sqrt(4*x+1)*(1+x)-3*x-1)/(sqrt(4*x+1)*(x^2+3*x+1)-4*x^2-5*x-1))) \\ G. C. Greubel, Mar 24 2017

G.f.: (1/2)*(sqrt(4*x+1)*(1+x)-3*x-1)/(sqrt(4*x+1)*(x^2+3*x+1)-4*x^2-5*x-1). - Vladimir Kruchinin, Mar 28 2016
a(n) ~ (-1)^(n+1) *2^(2*n+1) / (5*sqrt(Pi*n)). - Vaclav Kotesovec, Mar 28 2016
Conjecture: +2*n*a(n) +8*n*a(n-1) +(-n+20)*a(n-2) +5*(-n+4)*a(n-3) +2*(-2*n+5)*a(n-4)=0. - R. J. Mathar, Jun 14 2016

A360211 a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(2n-3k,n-2*k).

Original entry on oeis.org

1, 2, 5, 17, 61, 221, 812, 3021, 11344, 42899, 163146, 623320, 2390653, 9198879, 35494701, 137290466, 532149805, 2066501909, 8038146035, 31312535610, 122140123201, 477002869614, 1864912495716, 7298427590543, 28588888586743, 112080607196843, 439744801379594

Offset: 0

Seiichi Manyama, Jan 30 2023

nonn

Cf. A005317, A024718, A176332, A360185.
Cf. A000108, A176287.
Cf. A026641, A360212.

A360211 := proc(n)
    add((-1)^k*binomial(2*n-3*k,n-2*k),k=0..floor(n/2)) ;
end proc:
seq(A360211(n),n=0..40) ; # R. J. Mathar, Mar 02 2023

a(n) = sum(k=0, n\2, (-1)^k*binomial(2*n-3*k, n-2*k));

my(N=30, x='x+O('x^N)); Vec(1/(sqrt(1-4*x)*(1+2*x^2/(1+sqrt(1-4*x)))))

G.f.: 1 / ( sqrt(1-4*x) * (1 + x^2 * c(x)) ), where c(x) is the g.f. of A000108.
a(n) ~ 2^(2*n+3) / (9*sqrt(Pi*n)). - Vaclav Kotesovec, Feb 18 2023
D-finite with recurrence 2*n*a(n) +(-5*n+2)*a(n-1) +(-11*n+12)*a(n-2) +2*(-n+5)*a(n-3) +(-7*n+2)*a(n-4) +2*(-2*n+5)*a(n-5)=0. - R. J. Mathar, Mar 02 2023

cp's OEIS Frontend

A176335 Central coefficients T(2n,n) of number triangle A176331.

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A176331 Triangle read by rows: T(n, k) = Sum_{j=0..n} C(j, n-k) * C(j, k) * (-1)^(n - j).

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A176333 Expansion of (1-3*x)/(1-4*x+9*x^2).

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A176334 Diagonal sums of number triangle A176331.

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A181933 a(n) = Sum_{k=0..n} binomial(n+k,k)*sin(Pi*(n+k)/2).

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A360211 a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(2*n-3*k,n-2*k).

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A176333 Expansion of (1-3x)/(1-4x+9*x^2).

A181933 a(n) = Sum_{k=0..n} binomial(n+k,k)sin(Pi(n+k)/2).

A360211 a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(2n-3k,n-2*k).