{"id":"n-bf439299968f698d","revision":0,"first_index":0,"fingerprint":"f6-78c50abde97f5fa5","name":"one_plus_derivative of veronese_section_2_1 of Cube of lower triangular normalized binomial matrix.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[[],["1"],["6"],["28","1"],["120","12"],["495","92","1"],["1998","576","18"],["7938","3213","192","1"],["31104","16632","1584","24"],["120285","81648","11178","328","1"],["459270","384912","70956","3360","30"],["1732104","1756161","416988","28710","500","1"],["6456024","7794468","2309472","216216","6120","36"],["23796747","33776028","12196899","1482624","61425","708","1"],["86802030","143292240","61922718","9451728","535626","10080","42"]],"degrees":[-1,0,0,1,1,2,2,3,3,4,4,5,5,6,6],"row_sums":["0","1","6","29","132","588","2592","11344","49344","213440","918528","3934464","16782336","71314432","302014464"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["1"]],"recurrence":{"denominator":{"coeffs":[["-180","150","-12"],["195","-162","12"],["-18","12","0"],["3","0","0"]]},"terms":[{"coeff":{"coeffs":[["0","-44","92","-8"],["504","-418","-18","4"],["-27","29","4","0"],["9","1","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","360","-796","156","-8"],["0","-54","78","-8","0"],["0","18","-2","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":176412,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 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-65c43620d9797120"]},{"id":176413,"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-65c43620d9797120"]},{"id":176414,"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-65c43620d9797120"]}],"region_exact":{},"incoming_edges":[{"id":"e-65c43620d9797120","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-bf439299968f698d","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":"d612c0859efb1eb9cc9a6ba903fc9dc7cb507030e1fb82425575a9a5877b77b1","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-abf94b4038449932","role":"sequence","name":"veronese_section_2_1 of Cube of lower triangular normalized binomial matrix.","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:      0;\n  1:      1;\n  2:      6;\n  3:     28,     1;\n  4:    120,    12;\n  5:    495,    92,     1;\n  6:   1998,   576,    18;\n  7:   7938,  3213,   192,   1;\n  8:  31104, 16632,  1584,  24;\n  9: 120285, 81648, 11178, 328, 1;\n","recurrence_sentence":"The row polynomials satisfy (3*n^3-18*n^2+195*n-180+(12*n^2-162*n+150)*x+(12*n-12)*x^2)*P(n,x) = (9*n^3-27*n^2+504*n+(n^3+29*n^2-418*n-44)*x+(4*n^2-18*n+92)*x^2+(4*n-8)*x^3)*P(n-1,x) + ((18*n^2-54*n+360)*x-(2*n^2-78*n+796)*x^2-(8*n-156)*x^3-8*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((9*n^3 - 27*n^2 + 504*n + (n^3 + 29*n^2 - 418*n - 44)*x + (4*n^2 - 18*n + 92)*x^2 + (4*n - 8)*x^3)*P[n - 1, x] + ((18*n^2 - 54*n + 360)*x - (2*n^2 - 78*n + 796)*x^2 - (8*n - 156)*x^3 - 8*x^4)*D[P[n - 1, x], x])/(3*n^3 - 18*n^2 + 195*n - 180 + (12*n^2 - 162*n + 150)*x + (12*n - 12)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((9*n^3 - 27*n^2 + 504*n + (n^3 + 29*n^2 - 418*n - 44)*x + (4*n^2 - 18*n + 92)*x^2 + (4*n - 8)*x^3)*v[i - 1] + ((18*n^2 - 54*n + 360)*x - (2*n^2 - 78*n + 796)*x^2 - (8*n - 156)*x^3 - 8*x^4)*deriv(v[i - 1], x))/(3*n^3 - 18*n^2 + 195*n - 180 + (12*n^2 - 162*n + 150)*x + (12*n - 12)*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-8d887bfc40d036b9","steps":["Veronese 2 (j=1)","D+1"],"edges":["e-59cb5e4e13a9546c","e-65c43620d9797120"],"text":"A027465 \u2192 Veronese 2 (j=1) \u2192 D+1"},"headline":"A027465 \u2192 Veronese 2 (j=1) \u2192 D+1","chain":null}