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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":"85aa4730cdac6343f41335693998a2bb084f462d2478dd5bb063138dd419fb2f","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-56eee146d6109546","role":"sequence","name":"Triangle T(n,k) read by rows: T(n,k) = (n-k+1)*T(n-1,k-1) + (3*k-2)*k*T(n-1,k), initialized by T(n,1) = T(n,n) = 1.","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: 1,        2;\n  2: 1,       12,         12;\n  3: 1,       86,        210,         126;\n  4: 1,      672,       3120,        4350,         1902;\n  5: 1,     5354,      47388,      118260,       116076,        39852;\n  6: 1,    42804,     786372,     3149796,      5203896,      3890592,       1092924;\n  7: 1,   342398,   14330862,    86673330,    214868412,    262940616,     158370948,      37613880;\n  8: 1,  2739144,  279377112,  2498597874,   8800303590,  15744909168,   15286565700,    7703720856,   1583720640;\n  9: 1, 21913106, 5661483672, 76032527544, 369946845432, 911522953908, 1270609608132, 1019952259776, 441173374848, 79861657752;\n","recurrence_sentence":"The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (1-(6*n-10)*x+(3*n^2-27*n+29)*x^2+(9*n^2-45*n+38)*x^3+(9*n^2-33*n+24)*x^4+(3*n^2-9*n+6)*x^5)*P(n-1,x) + (7*x-(6*n-39)*x^2-(24*n-87)*x^3-(36*n-97)*x^4-(24*n-54)*x^5-(6*n-12)*x^6)*d/dx P(n-1,x) + (3*x^2+15*x^3+30*x^4+30*x^5+15*x^6+3*x^7)*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((1 - (6*n - 10)*x + (3*n^2 - 27*n + 29)*x^2 + (9*n^2 - 45*n + 38)*x^3 + (9*n^2 - 33*n + 24)*x^4 + (3*n^2 - 9*n + 6)*x^5)*P[n - 1, x] + (7*x - (6*n - 39)*x^2 - (24*n - 87)*x^3 - (36*n - 97)*x^4 - (24*n - 54)*x^5 - (6*n - 12)*x^6)*D[P[n - 1, x], x] + (3*x^2 + 15*x^3 + 30*x^4 + 30*x^5 + 15*x^6 + 3*x^7)*D[P[n - 1, x], {x, 2}])/(1 + 2*x + x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((1 - (6*n - 10)*x + (3*n^2 - 27*n + 29)*x^2 + (9*n^2 - 45*n + 38)*x^3 + (9*n^2 - 33*n + 24)*x^4 + (3*n^2 - 9*n + 6)*x^5)*v[i - 1] + (7*x - (6*n - 39)*x^2 - (24*n - 87)*x^3 - (36*n - 97)*x^4 - (24*n - 54)*x^5 - (6*n - 12)*x^6)*deriv(v[i - 1], x) + (3*x^2 + 15*x^3 + 30*x^4 + 30*x^5 + 15*x^6 + 3*x^7)*deriv(deriv(v[i - 1], x), x))/(1 + 2*x + x^2)); 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-56eee146d6109546","steps":["x \u21a6 x/(x+1)"],"edges":["e-910de3c970b04e15"],"text":"A166972 \u2192 x \u21a6 x/(x+1)"},"headline":"A166972 \u2192 x \u21a6 x/(x+1)","chain":null}