A085841 -id:A085841 - cp's OEIS frontend

A085840 Triangle read by rows: T(n,m) = 4^m * (2n+1)! / ( (2n - 2m + 1)! (2*m)! ), row n has n+1 terms.

1, 1, 12, 1, 40, 80, 1, 84, 560, 448, 1, 144, 2016, 5376, 2304, 1, 220, 5280, 29568, 42240, 11264, 1, 312, 11440, 109824, 329472, 292864, 53248, 1, 420, 21840, 320320, 1647360, 3075072, 1863680, 245760

Offset: 0

Gary W. Adamson, Jul 05 2003

Row n has the unsigned coefficients of a polynomial whose roots are 2*tan(Pi*k/(2n+1)) [for k = 1 to 2n].

Polynomial of row n = Sum_{m=0..n} (-1)^m T(n,m) x^(2n-2m).

			1
x^2 - 12
x^4 - 40x^2 + 80
x^6 - 84x^4 + 560x^2 - 448
x^8 - 144x^6 + 2016x^4 - 5376x^2 + 2304
x^10 - 220x^8 + 5280x^6 - 29568x^4 + 42240x^2 - 11264
Polynomial #4 has eight roots: 2 tan (Pi*k/9) for k=1 to 8.

Cf. A085841, A285043, A284044, A285045, A285046.

for n from 0 to 10 do lprint(seq(4^k*binomial(2*n + 1, 2*k), k = 0..n)) end do; # Peter Bala, Apr 10 2017

From Peter Bala, Apr 10 2017: (Start)
O.g.f.: (1 - (1 - 4*x)*t)/(1 - 2*(1 + 4*x)*t + (1 - 4*x)^2*t^2) = 1 + (1 + 12*x)*t + (1 + 40*x + 80*x^2)*t^2 + ....
n_th row polynomial R(n,x) = 1/2*( (1 + 2*sqrt(x))^(2*n+1) + (1 - 2*sqrt(x))^(2*n+1) ). These polynomials occur in the expansion cosh((2*n + 1)*arctanh(2*x)) = R(n,x^2)/(1 - 4*x^2)^(n+1/2). See A285043 - A285046.
For n >= 1, R(n,x) = (1 - 4*x)^n( U(n,(1 + 4*x)/(1 - 4*x)) - U(n-1,(1 + 4*x)/(1 - 4*x)) ), where U(n,x) is the n-th Chebyshev polynomial of the second kind. (End)

Edited by Don Reble, Nov 13 2005

A229032 Triangle T(n,k), 0 <= k <= n, read by rows, defined by T(n,k) = 4^k * C(n+1,2*k+1).

Original entry on oeis.org

1, 2, 0, 3, 4, 0, 4, 16, 0, 0, 5, 40, 16, 0, 0, 6, 80, 96, 0, 0, 0, 7, 140, 336, 64, 0, 0, 0, 8, 224, 896, 512, 0, 0, 0, 0, 9, 336, 2016, 2304, 256, 0, 0, 0, 0, 10, 480, 4032, 7680, 2560, 0, 0, 0, 0, 0, 11, 660, 7392, 21120, 14080, 1024, 0, 0, 0, 0, 0

Offset: 0

Rui Duarte and António Guedes de Oliveira, Sep 11 2013

Row n is the sum of the convolution of A089627(p,i) with A089627(n-p,i), for p=0,1,...,n.

			Triangle:
1
2, 0
3, 4, 0
4, 16, 0, 0
5, 40, 16, 0, 0
6, 80, 96, 0, 0, 0
7, 140, 336, 64, 0, 0, 0
8, 224, 896, 512, 0, 0, 0, 0
9, 336, 2016, 2304, 256, 0, 0, 0, 0
10, 480, 4032, 7680, 2560, 0, 0, 0, 0, 0
11, 660, 7392, 21120, 14080, 1024, 0, 0, 0, 0, 0

Rui Duarte and António Guedes de Oliveira, A Famous Identity of Hajós in Terms of Sets, Journal of Integer Sequences, Vol. 17 (2014), Article 14.9.1.

T(n,k) = 4^k * C(n+1, 2*k+1).
T(n,k) = sum(p=0..n, sum(i=0..k, C(p,i)*C(p-i, i)*C(n-p,k-i)*C(n-p-k+i, k-i))).
T(n,k) = A085841(n/2,k), if n is even.
T(n,k) = 2^k * A105070(n,k).

cp's OEIS Frontend

A085840 Triangle read by rows: T(n,m) = 4^m * (2n+1)! / ( (2n - 2m + 1)! (2*m)! ), row n has n+1 terms.

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A229032 Triangle T(n,k), 0 <= k <= n, read by rows, defined by T(n,k) = 4^k * C(n+1,2*k+1).

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cp's OEIS Frontend

A085840 Triangle read by rows: T(n,m) = 4^m * (2*n+1)! / ( (2*n - 2*m + 1)! * (2*m)! ), row n has n+1 terms.

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Author

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Programs

Maple

Formula

Extensions

A229032 Triangle T(n,k), 0 <= k <= n, read by rows, defined by T(n,k) = 4^k * C(n+1,2*k+1).

Views

Author

Keywords

Comments

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Formula

A085840 Triangle read by rows: T(n,m) = 4^m * (2n+1)! / ( (2n - 2m + 1)! (2*m)! ), row n has n+1 terms.