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0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-62e158c677cf5006"]},{"id":190893,"position":1,"claim":"one_signed_coefficients","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":0},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 0: the coefficients of P_n have one sign","region_exact":null,"edges":["e-62e158c677cf5006"]},{"id":190894,"position":2,"claim":"hadamard_kernel","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":0},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 0, are Hadamard kernels (preserve 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(f(k))_(k >= 0) is a multiplier sequence, so x^k -> f(k) x^k maps every real-rooted polynomial to a real-rooted polynomial or zero (Laguerre; see T. Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"certificate_hash":"a540354ff965989ddd975b1aba51be68dc3005df62e042589cf8da174bfa9b8d","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-a0dd3d6ef57da7a5","role":"sequence","name":"Triangle read by rows: T(n, k) = (6*n-6*k+1)*T(n-1, k-1) + (6*k-5)*T(n-1, k) + 6*T(n-2, k-1) with 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,        4;\n  2: 1,       80,           9;\n  3: 1,      636,        1431,           16;\n  4: 1,     4552,       38286,        18208,            25;\n  5: 1,    31988,      700902,      1246048,        199925,            36;\n  6: 1,   224064,    10972503,     47758208,      30479175,       2016576,            49;\n  7: 1,  1568620,   158702841,   1402549040,    2191482875,     634811364,      19215595,           64;\n  8: 1, 10980536,  2196749412,  35511761696,  110021876950,   79901463816,   11960080132,    175688576,         81;\n  9: 1, 76863972, 29641285584, 820913597184, 4483796778150, 6456667360536, 2514047891376, 210782475264, 1556495433, 100;\n","recurrence_sentence":"The row polynomials satisfy (900*n^2-1860*n+961)*P(n,x) = (291600*n^2-311040*n+90396)*x^2*P(n-4,x) + (129600*n^2-138240*n+44640)*x^3*d/dx P(n-4,x) - ((21600*n^2-23040*n-133176)*x-(583200*n^3-1756080*n^2+1316520*n-273420)*x^2)*P(n-3,x) + ((21600*n^2-23040*n+40920)*x^2+(259200*n^3-1205280*n^2+1046304*n-273048)*x^3)*d/dx P(n-3,x) + ((129600*n^2-138240*n+44640)*x^3-(129600*n^2-138240*n+44640)*x^4)*d^2/dx^2 P(n-3,x) + (4500*n^2-4800*n+155-(21600*n^3-84240*n^2-46296*n+221092)*x+(291600*n^4-1153440*n^3+1479384*n^2-649512*n+31713)*x^2)*P(n-2,x) - ((10800*n^2-11520*n-38688)*x-(21600*n^3+66960*n^2-67176*n-193316)*x^2-(129600*n^4-937440*n^3+1569024*n^2-811704*n+49972)*x^3)*d/dx P(n-2,x) - ((32400*n^2-34560*n-32364)*x^2-(129600*n^3-246240*n^2+126144*n-43896)*x^3+(129600*n^3-278640*n^2+160704*n-11532)*x^4)*d^2/dx^2 P(n-2,x) - (3600*n^2-2940*n-806-(21600*n^3-62640*n^2+60264*n-19220)*x)*P(n-1,x) + ((10800*n^2-16920*n+372)*x-(21600*n^3-41040*n^2+15624*n+3844)*x^2)*d/dx P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1 + 4*x, P(2,x) = 1 + 80*x + 9*x^2, P(3,x) = 1 + 636*x + 1431*x^2 + 16*x^3.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 4*x;\nP[2, x_] := 1 + 80*x + 9*x^2;\nP[3, x_] := 1 + 636*x + 1431*x^2 + 16*x^3;\nP[n_, x_] := P[n, x] = Expand[Together[((291600*n^2 - 311040*n + 90396)*x^2*P[n - 4, x] + (129600*n^2 - 138240*n + 44640)*x^3*D[P[n - 4, x], x] - ((21600*n^2 - 23040*n - 133176)*x - (583200*n^3 - 1756080*n^2 + 1316520*n - 273420)*x^2)*P[n - 3, x] + ((21600*n^2 - 23040*n + 40920)*x^2 + (259200*n^3 - 1205280*n^2 + 1046304*n - 273048)*x^3)*D[P[n - 3, x], x] + ((129600*n^2 - 138240*n + 44640)*x^3 - (129600*n^2 - 138240*n + 44640)*x^4)*D[P[n - 3, x], {x, 2}] + (4500*n^2 - 4800*n + 155 - (21600*n^3 - 84240*n^2 - 46296*n + 221092)*x + (291600*n^4 - 1153440*n^3 + 1479384*n^2 - 649512*n + 31713)*x^2)*P[n - 2, x] - ((10800*n^2 - 11520*n - 38688)*x - (21600*n^3 + 66960*n^2 - 67176*n - 193316)*x^2 - (129600*n^4 - 937440*n^3 + 1569024*n^2 - 811704*n + 49972)*x^3)*D[P[n - 2, x], x] - ((32400*n^2 - 34560*n - 32364)*x^2 - (129600*n^3 - 246240*n^2 + 126144*n - 43896)*x^3 + (129600*n^3 - 278640*n^2 + 160704*n - 11532)*x^4)*D[P[n - 2, x], {x, 2}] - (3600*n^2 - 2940*n - 806 - (21600*n^3 - 62640*n^2 + 60264*n - 19220)*x)*P[n - 1, x] + ((10800*n^2 - 16920*n + 372)*x - (21600*n^3 - 41040*n^2 + 15624*n + 3844)*x^2)*D[P[n - 1, x], x])/(900*n^2 - 1860*n + 961)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 4*x; v[3] = 1 + 80*x + 9*x^2; v[4] = 1 + 636*x + 1431*x^2 + 16*x^3; for(n = 4, N, my(i = n + 1); v[i] = ((291600*n^2 - 311040*n + 90396)*x^2*v[i - 4] + (129600*n^2 - 138240*n + 44640)*x^3*deriv(v[i - 4], x) - ((21600*n^2 - 23040*n - 133176)*x - (583200*n^3 - 1756080*n^2 + 1316520*n - 273420)*x^2)*v[i - 3] + ((21600*n^2 - 23040*n + 40920)*x^2 + (259200*n^3 - 1205280*n^2 + 1046304*n - 273048)*x^3)*deriv(v[i - 3], x) + ((129600*n^2 - 138240*n + 44640)*x^3 - (129600*n^2 - 138240*n + 44640)*x^4)*deriv(deriv(v[i - 3], x), x) + (4500*n^2 - 4800*n + 155 - (21600*n^3 - 84240*n^2 - 46296*n + 221092)*x + (291600*n^4 - 1153440*n^3 + 1479384*n^2 - 649512*n + 31713)*x^2)*v[i - 2] - ((10800*n^2 - 11520*n - 38688)*x - (21600*n^3 + 66960*n^2 - 67176*n - 193316)*x^2 - (129600*n^4 - 937440*n^3 + 1569024*n^2 - 811704*n + 49972)*x^3)*deriv(v[i - 2], x) - ((32400*n^2 - 34560*n - 32364)*x^2 - (129600*n^3 - 246240*n^2 + 126144*n - 43896)*x^3 + (129600*n^3 - 278640*n^2 + 160704*n - 11532)*x^4)*deriv(deriv(v[i - 2], x), x) - (3600*n^2 - 2940*n - 806 - (21600*n^3 - 62640*n^2 + 60264*n - 19220)*x)*v[i - 1] + ((10800*n^2 - 16920*n + 372)*x - (21600*n^3 - 41040*n^2 + 15624*n + 3844)*x^2)*deriv(v[i - 1], x))/(900*n^2 - 1860*n + 961)); 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-a0dd3d6ef57da7a5","steps":["(\u03b8+1)\u00b2"],"edges":["e-62e158c677cf5006"],"text":"A144443 \u2192 (\u03b8+1)\u00b2"},"headline":"A144443 \u2192 (\u03b8+1)\u00b2","chain":null}