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the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"fc75ed6a32e1a89c90bc80901760fe77d414eabe19b859f27a2475a7589944e1","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-4689fb1386d440ad","role":"sequence","name":"T(n,k) = binomial(n,k)*binomial(n+k,k), 0 <= k <= n, triangle read by rows.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:       1;\n   2:       3,         2;\n   3:      13,        24,          6;\n   4:      63,       242,        150,         20;\n   5:     321,      2320,       2680,        840,         70;\n   6:    1683,     21842,      42000,      23240,       4410,        252;\n   7:    8989,    204456,     616854,     536760,     174090,      22176,       924;\n   8:   48639,   1913186,    8742510,   11242812,    5546310,    1181796,    108108,     3432;\n   9:  265729,  17939616,  121380144,  222037200,  156400860,   49585536,   7471464,   514800,   12870;\n  10: 1462563, 168732962, 1664916000, 4223186000, 4083539460, 1807397592, 398918520, 44753280, 2406690, 48620;\n","recurrence_sentence":"The row polynomials satisfy -(n-1)*P(n,x) = (n-2)*P(n-2,x) - (6*n-9+(4*n-6)*x)*P(n-1,x) - (4*n-6)*x*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 3 + 2*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 3 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[((n - 2)*P[n - 2, x] - (6*n - 9 + (4*n - 6)*x)*P[n - 1, x] - (4*n - 6)*x*D[P[n - 1, x], x])/(-n + 1)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 3 + 2*x; for(n = 3, N, my(i = n); v[i] = ((n - 2)*v[i - 2] - (6*n - 9 + (4*n - 6)*x)*v[i - 1] - (4*n - 6)*x*deriv(v[i - 1], x))/(-n + 1)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-4689fb1386d440ad","steps":["basis A008277"],"edges":["e-79cfeb324d9d7d1e"],"text":"A063007 \u2192 basis A008277"},"headline":"A063007 \u2192 basis A008277","chain":null}