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(PARI) A188002(n, k) = my(s, X, Y, p, pi, pj); s = (n+k-2)^2*n\2; Y = 'y + O('y^(s+1)); X = 'x + O('x^(n+1)); p = prod(i=1, n+k-2, 1/(1-X*Y^(i^2))); sum(i=0, n, pi=polcoef(p, i); sum(j=i, n-i, pj=polcoef(p, j); sum(d=0, s, polcoef(pi, d)*polcoef(pj, d)) * (2-(i==j)) )); \\ Max Alekseyev, Sep 18 2023

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Table starts

.....1.....1......1......1......1......1......1.......1.......1.......1......1

.....2.....3......4......5......6......7......8.......9......10......11.....12

.....3.....4......5......8......9.....10.....11......12......15......16.....17

....12....17.....25.....34.....46.....57.....75......88.....108.....125....147

....25....45.....64.....88....117....159....216.....270.....333.....421....500

...106...176....286....421....636....862...1206....1587....2114....2698...3450

...376...644...1055...1696...2596...3796...5443....7674...10392...14198..18641

..1670..2983...5191...8373..13343..20224..30358...43750...62354...86173.118859

..7315.13675..24135..40681..66452.105208.160866..242128..354103..510107.717077

.35808.67487.122238.211234.354806.573982.907542.1393159.2104002.3099873

Table starts

.....1.....1......1......1......1......1......1.......1.......1.......1......1

.....2.....3......4......5......6......7......8.......9......10......11.....12

.....3.....4......5......8......9.....10.....11......12......15......16.....17

....12....17.....25.....34.....46.....57.....75......88.....108.....125....147

....25....45.....64.....88....117....159....216.....270.....333.....421....500

...106...176....286....421....636....862...1206....1587....2114....2698...3450

...376...644...1055...1696...2596...3796...5443....7674...10392...14198..18641

..1670..2983...5191...8373..13343..20224..30358...43750...62354...86173.118859

..7315.13675..24135..40681..66452.105208.160866..242128..354103..510107.717077

.35808.67487.122238.211234.354806.573982.907542.1393159.2104002.3099873

Cf. A188003 (n=3), A188004 (n=4), A188005 (n=5), A188006 (n=6), A188007 (n=7), A188008 (n=8), A187994 (k=1), A187993 (k=n), A187995 (k=2), A187996 (k=3), A187997 (k=4), A187998 (k=5), A187999 (k=6), A188000 (k=7), A188001 (k=8),.

R. H. Hardin , Mar 18 2011

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1, 1, 2, 1, 3, 3, 1, 4, 4, 12, 1, 5, 5, 17, 25, 1, 6, 8, 25, 45, 106, 1, 7, 9, 34, 64, 176, 376, 1, 8, 10, 46, 88, 286, 644, 1670, 1, 9, 11, 57, 117, 421, 1055, 2983, 7315, 1, 10, 12, 75, 159, 636, 1696, 5191, 13675, 35808, 1, 11, 15, 88, 216, 862, 2596, 8373, 24135, 67487, 176971

f[n_, nminusfE_, E_] := f[n, nminusfE, E] = Module[{a, fEminus , fEplus, f0}, If[E == 0, If[n == 0, 1, 0], a = 0; For[fEminus = 0, fEminus <= nminusfE, fEminus++, For[fEplus = 0, fEplus <= nminusfE - fEminus, fEplus++, f0 = nminusfE - fEminus - fEplus; a = a + f[n - E^2*fEminus + E^2*fEplus, f0, E - 1]]]; a]];

T[n_, k_] := T[n, k] = f[0, n, n + k - 2];

Table[Table[ T[n, d - n], {n, 1, d - 1}], {d, 2, 12}] // Flatten (* Jean-François Alcover, Aug 21 2023, after R. J. Mathar *)

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R. J. Mathar, <a href="/A188002/b188002_2.txt">Table of n, a(n) for n = 1..434</a> augmenting an earlier file of 188 elements by R. H. Hardin