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for an input row that is a polynomial the degree condition holds automatically","lean_route":"fTransform","lean_declarations":["RealRooted.splits_fPolynomial"],"certificate_hash":"0b928af7aa3a2bcc8c3ceac4d7544b74675aabd4a7e06aebfd86c084725142fc","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-b11b90d367f09b77","role":"sequence","name":"basis_map_A013609 of Triangle read by rows: T(n,k) are the coefficients of Charlier polynomials: A046716 transposed, for 0 <= k <= n.","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:      2,        4;\n   3:      5,       20,       19;\n   4:     16,       94,      176,       106;\n   5:     65,      492,     1346,      1588,       685;\n   6:    326,     2968,    10492,     18088,     15262,       5056;\n   7:   1957,    20600,    88181,    197184,    243591,     157960,      42079;\n   8:  13700,   162650,   810924,   2207042,   3548988,    3377582,    1763988,    390454;\n   9: 109601,  1442952,  8168100,  26024520,  51135374,   63540824,   48818164,  21223128,   4000441;\n  10: 986410, 14224332, 89797032, 326330080, 753456012, 1147571928, 1154126408, 739681152, 274322778, 44881660;\n","recurrence_sentence":"The row polynomials satisfy (1+x)*P(n,x) = -(n-2+(3*n-6)*x+(3*n-6)*x^2+(n-2)*x^3)*P(n-2,x) + (n+(2*n+2)*x+(n+2)*x^2)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + 4*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 2 + 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n - 2 + (3*n - 6)*x + (3*n - 6)*x^2 + (n - 2)*x^3)*P[n - 2, x] + (n + (2*n + 2)*x + (n + 2)*x^2)*P[n - 1, x])/(1 + x)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 4*x; for(n = 3, N, my(i = n); v[i] = (-(n - 2 + (3*n - 6)*x + (3*n - 6)*x^2 + (n - 2)*x^3)*v[i - 2] + (n + (2*n + 2)*x + (n + 2)*x^2)*v[i - 1])/(1 + x)); 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-3272d5bc697b9cb4","steps":["basis A013609","x \u21a6 x/(x+1)"],"edges":["e-9a46600ceb40809a","e-0c92c16cb077f231"],"text":"A094816 \u2192 basis A013609 \u2192 x \u21a6 x/(x+1)"},"headline":"A094816 \u2192 basis A013609 \u2192 x \u21a6 x/(x+1)","chain":null}