%I #49 May 05 2023 11:03:25

%S 1,1,4,1,5,10,1,6,15,20,1,7,20,35,35,1,8,25,50,70,56,1,9,30,65,105,

%T 126,84,1,10,35,80,140,196,210,120,1,11,40,95,175,266,336,330,165,1,

%U 12,45,110,210,336,462,540,495,220,1,13,50,125,245,406,588,750,825,715,286

%N Square array of 4D pyramidal numbers, read by antidiagonals.

%C The first row contains the tetrahedral numbers, which are really three-dimensional, but can be regarded as degenerate 4D pyramidal numbers. - _N. J. A. Sloane_, Aug 28 2015

%H <a href="/index/Ps#pyramidal_numbers">Index to sequences related to pyramidal numbers</a>

%F T(n, k) = binomial(k + 4, 4) + (n-1)*binomial(k + 3, 4), corrected Oct 01 2021.

%F T(n, k) = T(n - 1, k) + C(k + 3, 4) = T(n - 1, k) + k(k + 1)(k + 2)(k + 3)/24.

%F G.f. for rows: (1 + nx)/(1 - x)^5, n >= -1.

%F T(n,k) = sum_{j=0..k} A080851(n,j). - _R. J. Mathar_, Jul 28 2016

%e Array, n >= 0, k >= 0, begins

%e 1 4 10 20 35 56 ...

%e 1 5 15 35 70 126 ...

%e 1 6 20 50 105 196 ...

%e 1 7 25 65 140 266 ...

%e 1 8 30 80 175 336 ...

%p A080852 := proc(n,k)

%p binomial(k+4,4)+(n-1)*binomial(k+3,4) ;

%p end proc:

%p seq( seq(A080852(d-k,k),k=0..d),d=0..12) ; # _R. J. Mathar_, Oct 01 2021

%t T[n_, k_] := Binomial[k+3, 3] + Binomial[k+3, 4]n;

%t Table[T[n-k, k], {n, 0, 11}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, May 05 2023 *)

%o (Derive) vector(vector(poly_coeff(Taylor((1+kx)/(1-x)^5,x,11),x,n),n,0,11),k,-1,10)

%o (Derive) VECTOR(VECTOR(comb(k+3,3)+comb(k+3,4)n, k, 0, 11), n, 0, 11)

%Y Rows include A000292, A000332, A002415, A001296, A002418, A002419, A051740, A051797.

%Y Cf. A057145, A080851, A180266, A055796 (antidiagonal sums).

%Y See A257200 for another version of the array.

%K easy,nonn,tabl

%O 0,3

%A _Paul Barry_, Feb 21 2003