{"id":"n-aaf5f334c8903a15","revision":0,"first_index":0,"fingerprint":"f6-e468febdff774e05","name":"theta_plus_one_squared of Triangle T(n,k) read by rows: T(n,1) = T(n,n)=1, otherwise T(n,k) = (3n-3k+1)*T(n-1,k-1) + k*(3k-2)*T(n-1,k), 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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-bfb68eea1887527f"]},{"id":189623,"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-bfb68eea1887527f"]},{"id":189624,"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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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":"f3fc370284180ff17946f141f500a05628c729b043b7351d4fa22cfb26f71ae0","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-25ac1cc25c118491","role":"sequence","name":"Triangle T(n,k) read by rows: T(n,1) = T(n,n)=1, otherwise T(n,k) = (3n-3k+1)*T(n-1,k-1) + k*(3k-2)*T(n-1,k), 1<=k<=n.","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,        48,            9;\n  3: 1,       412,          621,            16;\n  4: 1,      3336,        19530,          5056,            25;\n  5: 1,     26740,       485190,        445280,         33225,            36;\n  6: 1,    213984,     10971135,      26436800,       7029875,        194832,           49;\n  7: 1,   1711948,    238097259,    1311027120,     870016875,      89565012,      1067269,          64;\n  8: 1,  13695672,   5073228216,   59213629056,   83181335250,   21126484152,    995302308,     5587200,       81;\n  9: 1, 109565476, 107215728300, 2539907537536, 6887127517650, 3585297074472, 419930130144, 10083254016, 28303425, 100;\n","recurrence_sentence":"The row polynomials satisfy (3*n-2)*P(n,x) = ((72*n^2-96*n-40)*x+(243*n^3-540*n^2+207*n+30)*x^2)*P(n-2,x) + ((48*n+16)*x+(171*n^2-183*n-44)*x^2+(108*n^3-594*n^2+297*n+34)*x^3)*d/dx P(n-2,x) + ((78*n+26)*x^2+(387*n^2-237*n-32)*x^3-(108*n^2-63*n-6)*x^4)*d^2/dx^2 P(n-2,x) + ((18*n+6)*x^3+(108*n^2-63*n-6)*x^4)*d^3/dx^3 P(n-2,x) + (3*n-2+(36*n^2-48*n+16)*x)*P(n-1,x) + ((15*n-16)*x-(36*n^2-12*n-8)*x^2)*d/dx P(n-1,x) + (9*n-6)*x^2*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[(((72*n^2 - 96*n - 40)*x + (243*n^3 - 540*n^2 + 207*n + 30)*x^2)*P[n - 2, x] + ((48*n + 16)*x + (171*n^2 - 183*n - 44)*x^2 + (108*n^3 - 594*n^2 + 297*n + 34)*x^3)*D[P[n - 2, x], x] + ((78*n + 26)*x^2 + (387*n^2 - 237*n - 32)*x^3 - (108*n^2 - 63*n - 6)*x^4)*D[P[n - 2, x], {x, 2}] + ((18*n + 6)*x^3 + (108*n^2 - 63*n - 6)*x^4)*D[P[n - 2, x], {x, 3}] + (3*n - 2 + (36*n^2 - 48*n + 16)*x)*P[n - 1, x] + ((15*n - 16)*x - (36*n^2 - 12*n - 8)*x^2)*D[P[n - 1, x], x] + (9*n - 6)*x^2*D[P[n - 1, x], {x, 2}])/(3*n - 2)]];\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; for(n = 2, N, my(i = n + 1); v[i] = (((72*n^2 - 96*n - 40)*x + (243*n^3 - 540*n^2 + 207*n + 30)*x^2)*v[i - 2] + ((48*n + 16)*x + (171*n^2 - 183*n - 44)*x^2 + (108*n^3 - 594*n^2 + 297*n + 34)*x^3)*deriv(v[i - 2], x) + ((78*n + 26)*x^2 + (387*n^2 - 237*n - 32)*x^3 - (108*n^2 - 63*n - 6)*x^4)*deriv(deriv(v[i - 2], x), x) + ((18*n + 6)*x^3 + (108*n^2 - 63*n - 6)*x^4)*deriv(deriv(deriv(v[i - 2], x), x), x) + (3*n - 2 + (36*n^2 - 48*n + 16)*x)*v[i - 1] + ((15*n - 16)*x - (36*n^2 - 12*n - 8)*x^2)*deriv(v[i - 1], x) + (9*n - 6)*x^2*deriv(deriv(v[i - 1], x), x))/(3*n - 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-25ac1cc25c118491","steps":["(\u03b8+1)\u00b2"],"edges":["e-bfb68eea1887527f"],"text":"A166962 \u2192 (\u03b8+1)\u00b2"},"headline":"A166962 \u2192 (\u03b8+1)\u00b2","chain":null}