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A091042 Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318. 24
1, 1, 3, 1, 10, 5, 1, 21, 35, 7, 1, 36, 126, 84, 9, 1, 55, 330, 462, 165, 11, 1, 78, 715, 1716, 1287, 286, 13, 1, 105, 1365, 5005, 6435, 3003, 455, 15, 1, 136, 2380, 12376, 24310, 19448, 6188, 680, 17, 1, 171, 3876, 27132, 75582, 92378, 50388, 11628, 969, 19, 1, 210, 5985, 54264, 203490, 352716, 293930, 116280, 20349, 1330, 21 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

The row polynomials Pe(n, x) := Sum_{m=0..n} a(n, m)*x^m appear as numerators of the generating functions for the even numbered column sequences of array A034870.

Elements have the same parity as those of Pascal's triangle.

All zeros of polynomial Pe(n, x) are negative. They are -tan^2(Pi/2*n+1), -tan^2(2*Pi/2*n+1), ..., -tan^2(n*Pi/2*n+1). Moreover, for m >= 1, Pe(m, -x^2) is the characteristic polynomial of the linear difference equation with constant coefficients for differences between multiples of 2*m+1 with even and odd digit sum in base 2*m in the interval [0,(2*m)^n). - Vladimir Shevelev and Peter J. C. Moses, May 22 2012

Row reverse of A103327. - Peter Bala, Jul 29 2013

The row polynomial Pe(d, x), multiplied by (2*d)!/d! = A001813(d), gives the numerator polynomial of the o.g.f. of the sequence of the diagonal d, for d >= 0, of the Sheffer triangle Lah[4,1] given in A048854. - Wolfdieter Lang, Oct 12 2017

REFERENCES

A. M. Yaglom and I. M. Yaglom, An elementary proof of the Wallis, Leibniz and Euler formulas for pi. Uspekhi Matem. Nauk, VIII (1953), 181-187(in Russian).

LINKS

Indranil Ghosh, Rows 0..120 of triangle, flattened

Wolfdieter Lang, First 9 rows.

V. Shevelev, On monotonic strengthening of Newman-like phenomenon on (2m+1)-multiples in base 2m, arXiv:0710.3177 [math.NT], 2007.

V. Shevelev and P. Moses, Tangent power sums and their applications, arXiv:1207.0404 [math.NT], 2012-2014.

FORMULA

T(n, m) = binomial(2*n+1, 2*m) = A007318(2*n+1, 2*m), n >= m >= 0, otherwise 0.

From Peter Bala, Jul 29 2013: (Start)

E.g.f.: sinh(t)*cosh(sqrt(x)*t) = t + (1 + 3*x)*t^3/3! + (1 + 10*x + 5*x^2)*t^5/5! + (1 + 21*x + 35*x^2 + 7*x^3)*t^7/7! + ....

O.g.f.: A(x,t) = (1 + (x - 1)*t)/( (1 + (x - 1)*t)^2 - 4*t*x ) = 1 + (1 + 3*x)*t + (1 + 10*x + 5*x^2)*t^2 + ...

The function A( x/(x + 4), t*(x + 4)/4 ) = 1 + (1 + x)*t + (1 + 3*x + x^2)*t^2 + ... is the o.g.f. for A085478.

O.g.f. for n-th diagonal: ( Sum_{k = 0..n} binomial(2*n,2*k)*x^k )/(1 - x)^(2*n).

n-th row polynomial R(n,x) = (1/2)*( (1 + sqrt(x))^(2*n+1) - (sqrt(x) - 1)^(2*n+1) ).

Row sums A000302. (End)

T(n, k) = 2*T(n-1,k) + 2*T(n-1,k-1) + 2*T(n-2,k-1) - T(n-2,k) - T(n-2,k-2) with T(0,0)=T(1,0)=1, T(1,1)=3, T(n,k)=0 if k < 0 or if k > n. - Philippe Deléham, Nov 26 2013

From Peter Bala, Jan 31 2022: (Start)

Define S(r,N) = Sum_{j = 1..N} j^r. Then the following identity holds for n >= 0: (1/2)*(N^2 + N)^(2*n+1) = T(n,0)*S(2*n+1,N) + T(n,1)*S(2*n+3,N) + ... + T(n,n)* S(4*n+1,N). Some examples are given below. (End)

EXAMPLE

Triangle a(n, m) begins:

n\m 0 1 2 3 4 5 6 7 8 9 10 ...

0: 1

1: 1 3

2: 1 10 5

3: 1 21 35 7

4: 1 36 126 84 9

5: 1 55 330 462 165 11

6: 1 78 715 1716 1287 286 13

7: 1 105 1365 5005 6435 3003 455 15

8: 1 136 2380 12376 24310 19448 6188 680 17

9: 1 171 3876 27132 75582 92378 50388 11628 969 19

10: 1 210 5985 54264 203490 352716 293930 116280 20349 1330 21

... reformatted and extended. - Wolfdieter Lang, Oct 12 2017

From Peter Bala, Jan 30 2022: (Start)

(1/2)*(N^2 + N) = Sum_{j = 1..N} j.

(1/2)*(N^2 + N)^3 = Sum_{j = 1..N} j^3 + 3*Sum_{j = 1..N} j^5.

(1/2)*(N^2 + N)^5 = Sum_{j = 1..N} j^5 + 10*Sum_{j = 1..N} j^7 + 5*Sum_{j = 1..N} j^9.

(1/2)*(N^2 + N)^7 = Sum_{j = 1..N} j^7 + 21*Sum_{j = 1..N} j^9 + 35*Sum_{j = 1..N} j^11 + 7*Sum_{j = 1..N} j^13. (End)

MAPLE

f := (x, t) -> cosh(sqrt(x)*t)*sinh(t); seq(seq(coeff(((2*n+1)!*coeff(series(f(x, t), t, 2*n+2), t, 2*n+1)), x, k), k=0..n), n=0..9); # Peter Luschny, Jul 29 2013

MATHEMATICA

T[n_, k_] /; 0 <= k <= n := T[n, k] = 2T[n-1, k] + 2T[n-1, k-1] + 2T[n-2, k-1] - T[n-2, k] - T[n-2, k-2]; T[0, 0] = T[1, 0] = 1; T[1, 1] = 3; T[_, _] = 0;

Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* Jean-François Alcover, Jul 29 2018, after Philippe Deléham *)

Table[Binomial[2*n+1, 2*k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Aug 01 2019 *)

PROG

(PARI) T(n, k) = binomial(2*n+1, 2*k); \\ G. C. Greubel, Aug 01 2019

(Magma) [[Binomial(2*n+1, 2*k): k in [0..n]]: n in [0..12]]; // G. C. Greubel, Aug 01 2019

(Sage) [[binomial(2*n+1, 2*k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Aug 01 2019

(GAP) Flat(List([0..12], n-> List([0..n], k-> Binomial(2*n+1, 2*k) ))); # G. C. Greubel, Aug 01 2019

CROSSREFS

Cf. A212500, A038754. A000302 (row sums), A085478, A103327 (row reverse), A048854, A103328.

Sequence in context: A107870 A078817 A316193 * A111418 A113187 A340554

Adjacent sequences: A091039 A091040 A091041 * A091043 A091044 A091045

KEYWORD

nonn,easy,tabl

AUTHOR

Wolfdieter Lang, Jan 23 2004

STATUS

approved

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Last modified December 2 12:14 EST 2022. Contains 358493 sequences. (Running on oeis4.)