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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-ce0dee53a3e61021"]},{"id":191282,"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-ce0dee53a3e61021"]},{"id":191283,"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 real-rootedness)","region_exact":null,"edges":["e-ce0dee53a3e61021"]},{"id":191284,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-ce0dee53a3e61021"]}],"region_exact":{},"incoming_edges":[{"id":"e-ce0dee53a3e61021","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-581672be31678c72","premise_count":1,"theorem":"Laguerre's theorem on differential operators: if phi is a real polynomial with only real roots, then phi(D) maps every real-rooted polynomial to a real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"0ea0f966139fdee64bc284fb04ce0af1895d9c7731fedefd8502712894eea411","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-2d05bfc763190112","role":"sequence","name":"Triangle, read by rows, T(n,k) = (7*n-7*k+1)*T(n-1, k-2) + (7*k-6)*T(n-1, k) + 7*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:        2,           1;\n  2:       24,          25,            1;\n  3:      207,         618,          209,             1;\n  4:     1678,       14359,        11372,          1681,             1;\n  5:    13453,      280354,       533804,        187259,         13457,             1;\n  6:   107660,     4914883,     20152305,      15530679,       2941907,        107665,            1;\n  7:   861323,    80909458,    642412029,    1003979935,     400916327,      45192000,       861329,           1;\n  8:  6890634,  1283193591,  18263761817,   52625526898,   41063598093,    9704112320,    686385910,     6890641,        1;\n  9: 55125129, 19900791306, 480848306744, 2364386121853, 3311116445952, 1493703688302, 226434989508, 10363834113, 55125137, 1;\n","recurrence_sentence":"The row polynomials satisfy 56*P(n,x) = (2058*n^2-6958*n+5432+(1029*n^2-3479*n-1400)*x-4802*x^2-(7203*n^2-31556*n+29988)*x^3)*P(n-4,x) + ((1029*n^2-3479*n+2716)*x-2058*x^2-(7203*n^2-31556*n+29988)*x^3)*d/dx P(n-4,x) + (2058*n^3-14749*n^2+31773*n-20564+(1029*n^3-4361*n^2-2534*n+24230)*x-(1029*n^2+3038*n-18648)*x^2-(7203*n^3-52136*n^2+120148*n-80192)*x^3)*P(n-3,x) + (2205*n^2-7455*n+5820+(1029*n^3-9506*n^2+23093*n-20318)*x-(2058*n^2-5929*n-5054)*x^2-(7203*n^3-59339*n^2+151704*n-117726)*x^3+(7203*n^2-31556*n+29988)*x^4)*d/dx P(n-3,x) + ((1029*n^2-3479*n+2716)*x-(1029*n^2-3479*n+4774)*x^2-(7203*n^2-31556*n+27930)*x^3+(7203*n^2-31556*n+29988)*x^4)*d^2/dx^2 P(n-3,x) - (147*n^2+1855*n-2748+(4116*n^2-24794*n+27930)*x+(1715*n^2-9800*n+13608)*x^2+(5488*n-10192)*x^3)*P(n-2,x) - (147*n^2-497*n+388+(2058*n-8918)*x-(1029*n^2-2450*n-8456)*x^2+1372*x^3-5488*x^4)*d/dx P(n-2,x) + (2744*x^2-5488*x^3+2744*x^4)*d^2/dx^2 P(n-2,x) + (294*n+56+(784*n-2828)*x+784*x^2)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2 + x, P(2,x) = 24 + 25*x + x^2, P(3,x) = 207 + 618*x + 209*x^2 + x^3.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + x;\nP[2, x_] := 24 + 25*x + x^2;\nP[3, x_] := 207 + 618*x + 209*x^2 + x^3;\nP[n_, x_] := P[n, x] = Expand[Together[((2058*n^2 - 6958*n + 5432 + (1029*n^2 - 3479*n - 1400)*x - 4802*x^2 - (7203*n^2 - 31556*n + 29988)*x^3)*P[n - 4, x] + ((1029*n^2 - 3479*n + 2716)*x - 2058*x^2 - (7203*n^2 - 31556*n + 29988)*x^3)*D[P[n - 4, x], x] + (2058*n^3 - 14749*n^2 + 31773*n - 20564 + (1029*n^3 - 4361*n^2 - 2534*n + 24230)*x - (1029*n^2 + 3038*n - 18648)*x^2 - (7203*n^3 - 52136*n^2 + 120148*n - 80192)*x^3)*P[n - 3, x] + (2205*n^2 - 7455*n + 5820 + (1029*n^3 - 9506*n^2 + 23093*n - 20318)*x - (2058*n^2 - 5929*n - 5054)*x^2 - (7203*n^3 - 59339*n^2 + 151704*n - 117726)*x^3 + (7203*n^2 - 31556*n + 29988)*x^4)*D[P[n - 3, x], x] + ((1029*n^2 - 3479*n + 2716)*x - (1029*n^2 - 3479*n + 4774)*x^2 - (7203*n^2 - 31556*n + 27930)*x^3 + (7203*n^2 - 31556*n + 29988)*x^4)*D[P[n - 3, x], {x, 2}] - (147*n^2 + 1855*n - 2748 + (4116*n^2 - 24794*n + 27930)*x + (1715*n^2 - 9800*n + 13608)*x^2 + (5488*n - 10192)*x^3)*P[n - 2, x] - (147*n^2 - 497*n + 388 + (2058*n - 8918)*x - (1029*n^2 - 2450*n - 8456)*x^2 + 1372*x^3 - 5488*x^4)*D[P[n - 2, x], x] + (2744*x^2 - 5488*x^3 + 2744*x^4)*D[P[n - 2, x], {x, 2}] + (294*n + 56 + (784*n - 2828)*x + 784*x^2)*P[n - 1, x])/(56)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + x; v[3] = 24 + 25*x + x^2; v[4] = 207 + 618*x + 209*x^2 + x^3; for(n = 4, N, my(i = n + 1); v[i] = ((2058*n^2 - 6958*n + 5432 + (1029*n^2 - 3479*n - 1400)*x - 4802*x^2 - (7203*n^2 - 31556*n + 29988)*x^3)*v[i - 4] + ((1029*n^2 - 3479*n + 2716)*x - 2058*x^2 - (7203*n^2 - 31556*n + 29988)*x^3)*deriv(v[i - 4], x) + (2058*n^3 - 14749*n^2 + 31773*n - 20564 + (1029*n^3 - 4361*n^2 - 2534*n + 24230)*x - (1029*n^2 + 3038*n - 18648)*x^2 - (7203*n^3 - 52136*n^2 + 120148*n - 80192)*x^3)*v[i - 3] + (2205*n^2 - 7455*n + 5820 + (1029*n^3 - 9506*n^2 + 23093*n - 20318)*x - (2058*n^2 - 5929*n - 5054)*x^2 - (7203*n^3 - 59339*n^2 + 151704*n - 117726)*x^3 + (7203*n^2 - 31556*n + 29988)*x^4)*deriv(v[i - 3], x) + ((1029*n^2 - 3479*n + 2716)*x - (1029*n^2 - 3479*n + 4774)*x^2 - (7203*n^2 - 31556*n + 27930)*x^3 + (7203*n^2 - 31556*n + 29988)*x^4)*deriv(deriv(v[i - 3], x), x) - (147*n^2 + 1855*n - 2748 + (4116*n^2 - 24794*n + 27930)*x + (1715*n^2 - 9800*n + 13608)*x^2 + (5488*n - 10192)*x^3)*v[i - 2] - (147*n^2 - 497*n + 388 + (2058*n - 8918)*x - (1029*n^2 - 2450*n - 8456)*x^2 + 1372*x^3 - 5488*x^4)*deriv(v[i - 2], x) + (2744*x^2 - 5488*x^3 + 2744*x^4)*deriv(deriv(v[i - 2], x), x) + (294*n + 56 + (784*n - 2828)*x + 784*x^2)*v[i - 1])/(56)); 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-2d05bfc763190112","steps":["D+1"],"edges":["e-ce0dee53a3e61021"],"text":"A144445 \u2192 D+1"},"headline":"A144445 \u2192 D+1","chain":null}