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of veronese_section_2_0 of Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-702b8764f2c773f5","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["5"],["30","15"]],"card_rows":[["1"],["5"],["30","15"],["140","105"]],"row_sums_preview":["1","5","45","245","1190","5166","20790","78870"],"regions":["(-inf, -1]","(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-79986ab5023c2764","steps":["Veronese 2","x \u21a6 x+1"],"edges":["e-8ba085c2fb60bb1b","e-68de572e63ababc1"],"text":"A178821 \u2192 Veronese 2 \u2192 x \u21a6 x+1"},"headline":"A178821 \u2192 Veronese 2 \u2192 x \u21a6 x+1","chain":null}}],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:      1;\n  1:      6;\n  2:     42,      21;\n  3:    224,     168;\n  4:   1008,    1008,    126;\n  5:   4032,    5040,   1260;\n  6:  14784,   22176,   8316,    462;\n  7:  50688,   88704,  44352,   5544;\n  8: 164736,  329472, 205920,  41184,  1287;\n  9: 512512, 1153152, 864864, 240240, 18018;\n","recurrence_sentence":"The row polynomials satisfy (n^2-n)*P(n,x) = (2*n^2+8*n-10+(n^2+4*n-5)*x)*P(n-1,x) - ((2*n+10)*x+(2*n+10)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 6.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 6;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n^2 + 8*n - 10 + (n^2 + 4*n - 5)*x)*P[n - 1, x] - ((2*n + 10)*x + (2*n + 10)*x^2)*D[P[n - 1, x], x])/(n^2 - n)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 6; for(n = 2, N, my(i = n + 1); v[i] = ((2*n^2 + 8*n - 10 + (n^2 + 4*n - 5)*x)*v[i - 1] - ((2*n + 10)*x + (2*n + 10)*x^2)*deriv(v[i - 1], x))/(n^2 - n)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-97228a88c81561dc","steps":["Veronese 2","x \u21a6 x+1"],"edges":["e-496b1966a8762df1","e-3dd245e74534b1c0"],"text":"A178822 \u2192 Veronese 2 \u2192 x \u21a6 x+1"},"headline":"A178822 \u2192 Veronese 2 \u2192 x \u21a6 x+1","chain":null}