{"id":"n-89eea40781af2171","revision":0,"first_index":0,"fingerprint":"f6-16c35582abfc142d","name":"derivative of veronese_section_2_0 of Triangle read by rows: triangle of repeated stepped binomial coefficients.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":-2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[[],[],["1"],["1"],["1","2"],["0","4"],["0","6","3"],["0","2","9"],["0","2","18","4"],["0","0","12","16"],["0","0","15","40","5"],["0","0","3","40","25"],["0","0","3","60","75","6"],["0","0","0","24","100","36"],["0","0","0","28","175","126","7"]],"degrees":[-1,-1,0,0,1,1,2,2,3,3,4,4,5,5,6],"row_sums":["0","0","1","1","3","4","9","11","24","28","60","68","144","160","336"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["0"],["1"],["1"],["1","2"],["0","4"]],"recurrence":{"denominator":{"coeffs":[["4"]]},"terms":[{"coeff":{"coeffs":[["-2","10","-18","4"]]},"deriv_order":0,"offset":6,"sign":"none"},{"coeff":{"coeffs":[["0","-1","5","-4"]]},"deriv_order":1,"offset":6,"sign":"none"},{"coeff":{"coeffs":[["-4","16","-12"]]},"deriv_order":0,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["0","-2","8"]]},"deriv_order":1,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["2","12"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1","-4"]]},"deriv_order":1,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":170929,"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-2722722c4727c67a"]},{"id":170930,"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-2722722c4727c67a"]},{"id":170931,"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-2722722c4727c67a"]}],"region_exact":{},"incoming_edges":[{"id":"e-2722722c4727c67a","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"0\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-89eea40781af2171","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":"b8164a0ad00f5f9fff5311a587f3799aaed1a74414aa5623673f62f817c4d359","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-aed87ddc8fcd289f","role":"sequence","name":"veronese_section_2_0 of Triangle read by rows: triangle of repeated stepped binomial coefficients.","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: 0;\n  2: 1;\n  3: 1;\n  4: 1, 2;\n  5: 0, 4;\n  6: 0, 6,  3;\n  7: 0, 2,  9;\n  8: 0, 2, 18,  4;\n  9: 0, 0, 12, 16;\n","recurrence_sentence":"The row polynomials satisfy 4*P(n,x) = -(2-10*x+18*x^2-4*x^3)*P(n-6,x) - (x-5*x^2+4*x^3)*d/dx P(n-6,x) - (4-16*x+12*x^2)*P(n-4,x) - (2*x-8*x^2)*d/dx P(n-4,x) + (2+12*x)*P(n-2,x) + (1-4*x)*d/dx P(n-2,x) for n >= 6, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 1, P(3,x) = 1, P(4,x) = 1 + 2*x, P(5,x) = 4*x.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 0;\nP[2, x_] := 1;\nP[3, x_] := 1;\nP[4, x_] := 1 + 2*x;\nP[5, x_] := 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(2 - 10*x + 18*x^2 - 4*x^3)*P[n - 6, x] - (x - 5*x^2 + 4*x^3)*D[P[n - 6, x], x] - (4 - 16*x + 12*x^2)*P[n - 4, x] - (2*x - 8*x^2)*D[P[n - 4, x], x] + (2 + 12*x)*P[n - 2, x] + (1 - 4*x)*D[P[n - 2, x], x])/(4)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 0; v[3] = 1; v[4] = 1; v[5] = 1 + 2*x; v[6] = 4*x; for(n = 6, N, my(i = n + 1); v[i] = (-(2 - 10*x + 18*x^2 - 4*x^3)*v[i - 6] - (x - 5*x^2 + 4*x^3)*deriv(v[i - 6], x) - (4 - 16*x + 12*x^2)*v[i - 4] - (2*x - 8*x^2)*deriv(v[i - 4], x) + (2 + 12*x)*v[i - 2] + (1 - 4*x)*deriv(v[i - 2], x))/(4)); 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-35bd700768cac55f","steps":["Veronese 2","D"],"edges":["e-f209261f0addbaea","e-2722722c4727c67a"],"text":"A103633 \u2192 Veronese 2 \u2192 D"},"headline":"A103633 \u2192 Veronese 2 \u2192 D","chain":null}