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the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"ec1bd87d90ed94f7b699d568ebddf495b6ed7d7d7a6a795253495ec7a7fb1052","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-13e23eca5f474574","role":"sequence","name":"Triangle of scaled 1-tiered binomial coefficients, T(n,k) = 2^(n+1)*(n-k,k)_1 (n >= 0, 0 <= k <= n), where (N,M)_1 is the 1-tiered binomial coefficient.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, 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":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:      4,        3;\n  2:     16,       29,        7;\n  3:     64,      215,      121,        15;\n  4:    256,     1445,     1435,       404,        31;\n  5:   1024,     9263,    14545,      7007,      1198,       63;\n  6:   4096,    57869,   135667,     99820,     28595,     3303,      127;\n  7:  16384,   356135,  1204441,   1273615,    544670,   103812,     8671,     255;\n  8:  65536,  2171765, 10361995,  15168804,   9098691,  2547258,   347004,   21974,   511;\n  9: 262144, 13168223, 87298465, 172545087, 139638198, 53906265, 10656282, 1090562, 54220, 1023;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(8+10*x+2*x^2)*P(n-2,x) - (10*x+4*x^2)*d/dx P(n-2,x) - 2*x^2*d^2/dx^2 P(n-2,x) + (6+3*x)*P(n-1,x) + 3*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 3*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4 + 3*x;\nP[n_, x_] := P[n, x] = Expand[-(8 + 10*x + 2*x^2)*P[n - 2, x] - (10*x + 4*x^2)*D[P[n - 2, x], x] - 2*x^2*D[P[n - 2, x], {x, 2}] + (6 + 3*x)*P[n - 1, x] + 3*x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = -(8 + 10*x + 2*x^2)*v[i - 2] - (10*x + 4*x^2)*deriv(v[i - 2], x) - 2*x^2*deriv(deriv(v[i - 2], x), x) + (6 + 3*x)*v[i - 1] + 3*x*deriv(v[i - 1], x)); 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-13e23eca5f474574","steps":["basis A008277"],"edges":["e-7c5991ee94b58083"],"text":"A308737 \u2192 basis A008277"},"headline":"A308737 \u2192 basis A008277","chain":null}