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A142978 Table of figurate numbers for the n-dimensional cross polytopes. 14
1, 1, 2, 1, 4, 3, 1, 6, 9, 4, 1, 8, 19, 16, 5, 1, 10, 33, 44, 25, 6, 1, 12, 51, 96, 85, 36, 7, 1, 14, 73, 180, 225, 146, 49, 8, 1, 16, 99, 304, 501, 456, 231, 64, 9, 1, 18, 129, 476, 985, 1182, 833, 344, 81, 10 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

The n-th row entries for this array are the regular polytope numbers for the n-dimensional cross polytope as defined by [Kim]. The rows are the partial sums of the rows of the square array of Delannoy numbers A008288.

The odd numbered rows of this array form A142977. For a triangular version of this table see A104698. Cf. also A101603.

The n-th row of the array is the binomial transform of n-th row of triangle A081277, followed by zeros. Example: row 4 (1, 6, 19, 44, 85,...) = b.t. of row 3 of A081277: (1, 5, 8, 4, 0, 0, 0,...). - Gary W. Adamson, Jul 17 2008

The main diagonal of the array T(n,k) is A047781 Sum(binomial(n-1,k)*binomial(n+k,k),k=0..n-1). Also a(n)=T(n,n), array T as in A049600. The link from A099193 to J. V. Post, Table of polytope numbers, Sorted, Through 1,000,000, includes all n-D Hyperoctahedron (n-Cross Polytope) Numbers through 10-Cross(20) = 1669752016. - Jonathan Vos Post, Jul 16 2008

REFERENCES

Steven Edwards and W. Griffiths, Generalizations of Delannoy and cross polytope numbers, Fib. Q., 55 (2017), 356-366.

LINKS

Reinhard Zumkeller, Rows n = 1..100 of triangle, flattened

M. Janjic and B. Petkovic, A Counting Function, arXiv 1301.4550 [math.CO], 2013.

M. Janjic, B. Petkovic, A Counting Function Generalizing Binomial Coefficients and Some Other Classes of Integers, J. Int. Seq. 17 (2014) # 14.3.5

Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., 131 (2002), 65-75.

Eric W. Weisstein, Meixner polynomial of the first kind

Eric W. Weisstein, Mittag-Leffler polynomial

Yutong Yang, From Simplest Recursion to the Recursion of Generalizations of Cross Polytope Numbers, (2017), Honors College Capstone and Theses, 13.

FORMULA

T(n,k) = sum {i = 0..n-1} C(n-1,i)*C(k+i,n).

Reciprocity law: n*T(n,k) = k*T(k,n).

Recurrence relation: T(n,1) = 1, T(1,k) = k, T(n,k) = T(n,k-1) + T(n-1,k-1) + T(n-1,k), n,k > 1.

O.g.f. row n: x*(1+x)^(n-1)/(1-x)^(n+1).

O.g.f. for array: sum_{n >= 1, k >= 1} T(n, k)*x^k*y^n = x*y/((1-x)*(1-x-y-x*y)).

The n-th row entries are the values [p_n(k)]k>=1 of the polynomial function p_n(x) = sum {k = 1..n} 2^(k-1)*C(n-1,k-1)*C(x,k). The first few values are p_1(x) = x, p_2(x) = x^2, p_3(x) = (2*x^3+x)/3 and p_4(x) = (x^4+2*x^2)/3. The polynomial p_n(x) is the unique polynomial solution of the difference equation x*(f(x+1)-f(x-1)) = 2*n*f(x), normalized so that f(1) = 1.

The o.g.f. for the p_n(x) is 1/2*((1+t)/(1-t))^x = 1/2 + x*t + x^2*t^2 + (2*x^3+x)/3*t^3 + ... . Thus p_n(x) is, apart from a constant factor, the Meixner polynomial of the first kind M_n(x;b,c) at b = 0, c = -1, also known as a Mittag-Leffler polynomial.

The entries in the n-th row appear in the series acceleration formula for the constant log(2): sum {k = 1..inf} (-1)^(k+1)/(T(n,k)*T(n,k+1)) = 1 + (-1)^(n+1) * (2*n)*[log(2) - (1-1/2+1/3- ...+ (-1)^(n+1)/n)]. For example, n = 3 gives log(2) = 4/6 + 1/6*(1/(1*6)-1/(6*19)+1/(19*44)-1/(44*85)+...). See A142983 for further details.

Contribution from Peter Bala, Oct 02 2008: (Start)

The odd-indexed columns of this array form the array A142992 of crystal ball sequences for lattices of type C_n. Conjectural congruences for main diagonal entries: Put A(n) = T(n,n). Calculation suggests the following congruences: for prime p > 3 and m,r >=1, A(m*p^r) = A(m*p^(r-1)) (mod p^(3*r));

sum {k = 0..p-1} A(k)^2 = 0 (mod p) if p is a prime of the form 8*n+1 or 8*n+7;

sum {k = 0..p-1} A(k)^2 = -1 (mod p) if p is a prime of the form 8*n+3 or 8*n+5;

(End)

EXAMPLE

The square array begins

n\k|1...2....3.....4.....5.......6

----------------------------------

.1.|1...2....3.....4......5......6... A000027

.2.|1...4....9....16.....25.....36... A000290

.3.|1...6...19....44.....85....146... A005900

.4.|1...8...33....96....225....456... A014820

.5.|1..10...51...180....501...1182... A069038

.6.|1..12...73...304....985...2668... A069039

.7.|1..14...99...476...1765...5418... A099193

...

MAPLE

with(combinat): T:=(n, k) -> add(binomial(n-1, i)*binomial(k+i, n), i = 0..n-1); for n from 1 to 10 do seq(T(n, k), k = 1..10) end do;

MATHEMATICA

t[n_, k_] := Sum[ Binomial[n-1, i]*Binomial[k+i, n], {i, 0, n-1}]; Table[t[n-k, k], {n, 1, 11}, {k, 1, n-1}] // Flatten (* Jean-Fran├žois Alcover, Mar 06 2013 *)

PROG

(Haskell)

a142978 n k = a142978_tabl !! (n-1) !! (k-1)

a142978_row n = a142978_tabl !! (n-1)

a142978_tabl = map reverse a104698_tabl

-- Reinhard Zumkeller, Jul 17 2015

CROSSREFS

Cf. A008288 (Delannoy numbers), A005900 (row 3), A014820 (row 4), A069038 (row 5), A069039 (row 6), A099193 (row 7), A099195 (row 8), A099196 (row 9), A099197 (row 10), A101603, A104698 (triangle version), A142977, A142983.

Cf. A049600, A047781, A081277.

Sequence in context: A180803 A093966 A103406 * A152060 A093190 A132191

Adjacent sequences:  A142975 A142976 A142977 * A142979 A142980 A142981

KEYWORD

easy,nonn,tabl

AUTHOR

Peter Bala, Jul 15 2008

EXTENSIONS

Restored missing program. - Peter Bala, Oct 02 2008

STATUS

approved

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Last modified February 21 18:45 EST 2019. Contains 320376 sequences. (Running on oeis4.)