{"id":"n-20dc22ed0b62a343","revision":0,"first_index":1,"fingerprint":"f6-f204ef93c1fbf845","name":"one_plus_derivative of derivative of basis-map coefficients of A scaled Hermite triangle.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["2","2"],["0","6","3"],["0","0","12","4"],["0","0","0","20","5"],["0","0","0","0","30","6"],["0","0","0","0","0","42","7"],["0","0","0","0","0","0","56","8"],["0","0","0","0","0","0","0","72","9"],["0","0","0","0","0","0","0","0","90","10"],["0","0","0","0","0","0","0","0","0","110","11"],["0","0","0","0","0","0","0","0","0","0","132","12"],["0","0","0","0","0","0","0","0","0","0","0","156","13"],["0","0","0","0","0","0","0","0","0","0","0","0","182","14"],["0","0","0","0","0","0","0","0","0","0","0","0","0","210","15"],["0","0","0","0","0","0","0","0","0","0","0","0","0","0","240","16"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["4","9","16","25","36","49","64","81","100","121","144","169","196","225","256"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["2","2"]],"recurrence":{"denominator":{"coeffs":[["0","0"],["-1","1"],["1","0"]]},"terms":[{"coeff":{"coeffs":[["0","0","1"],["0","1","1"],["0","1","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":171059,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-2eb130fced27a43a"]},{"id":171060,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-2eb130fced27a43a"]},{"id":171061,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-2eb130fced27a43a"]}],"region_exact":{},"incoming_edges":[{"id":"e-2eb130fced27a43a","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-20dc22ed0b62a343","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":"5baac3f9b6fd9e7b821dbd972b4ed2a305cd1017cd8b59b568cc173915641f15","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-e177181dad825c00","role":"sequence","name":"derivative of basis-map coefficients of A scaled Hermite triangle.","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: 2, 2;\n   2: 0, 6,  3;\n   3: 0, 0, 12,  4;\n   4: 0, 0,  0, 20,  5;\n   5: 0, 0,  0,  0, 30,  6;\n   6: 0, 0,  0,  0,  0, 42,  7;\n   7: 0, 0,  0,  0,  0,  0, 56,  8;\n   8: 0, 0,  0,  0,  0,  0,  0, 72,  9;\n   9: 0, 0,  0,  0,  0,  0,  0,  0, 90,  10;\n  10: 0, 0,  0,  0,  0,  0,  0,  0,  0, 110, 11;\n","recurrence_sentence":"The row polynomials satisfy (n^2-n+n*x)*P(n,x) = ((n^2+n)*x+(n+1)*x^2)*P(n-1,x) for n >= 2, with P(1,x) = 2 + 2*x.","mathematica":"P[1, x_] := 2 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[(((n^2 + n)*x + (n + 1)*x^2)*P[n - 1, x])/(n^2 - n + n*x)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 2 + 2*x; for(n = 2, N, my(i = n); v[i] = (((n^2 + n)*x + (n + 1)*x^2)*v[i - 1])/(n^2 - n + n*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-5fe240122e98ffb3","steps":["D","D+1"],"edges":["e-35fb5901b32bd10b","e-2eb130fced27a43a"],"text":"basis-map coefficients of A112227 \u2192 D \u2192 D+1"},"headline":"basis-map coefficients of A112227 \u2192 D \u2192 D+1","chain":null}