{"id":"n-a4fde56ab7166190","revision":0,"first_index":0,"fingerprint":"f6-332478c01a476533","name":"derivative of veronese_section_2_0 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":-2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[[],[],["1"],["9"],["54","2"],["270","30"],["1215","270","3"],["5103","1890","63"],["20412","11340","756","4"],["78732","61236","6804","108"],["295245","306180","51030","1620","5"],["1082565","1443420","336798","17820","165"],["3897234","6495390","2020788","160380","2970","6"],["13817466","28146690","11258676","1250964","38610","234"],["48361131","118216098","59108049","8756748","405405","4914","7"]],"degrees":[-1,-1,0,0,1,1,2,2,3,3,4,4,5,5,6],"row_sums":["0","0","1","9","56","300","1488","7056","32512","146880","654080","2880768","12576768","54512640","234852352"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["0"],["1"]],"recurrence":{"denominator":{"coeffs":[["6"],["-9"],["3"]]},"terms":[{"coeff":{"coeffs":[["0","0"],["-9","-3"],["9","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","0"],["0","18","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":176135,"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-7375e7ec9839fed4"]},{"id":176136,"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-7375e7ec9839fed4"]},{"id":176137,"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-7375e7ec9839fed4"]}],"region_exact":{},"incoming_edges":[{"id":"e-7375e7ec9839fed4","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-a4fde56ab7166190","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":"841723beef154c143ccfff162229b1ddd495c732ec49f576e7aa87487acdd4e4","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-081b3f700db2d097","role":"sequence","name":"veronese_section_2_0 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":[{"id":"l-9c294e423db45bb8","kind":"fingerprint_related","node_a_id":"n-abf94b4038449932","node_b_id":"n-a4fde56ab7166190","fingerprint":"f6-332478c01a476533","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-332478c01a476533"},"other_id":"n-abf94b4038449932","other":{"id":"n-abf94b4038449932","name":"veronese_section_2_1 of Cube of lower triangular normalized binomial matrix.","first_index":0,"fingerprint":"f6-332478c01a476533","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["1"],["6"]],"card_rows":[["1"],["6"],["27","1"],["108","12"]],"row_sums_preview":["0","1","6","28","120","496","2016","8128"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8d887bfc40d036b9","steps":["Veronese 2 (j=1)"],"edges":["e-59cb5e4e13a9546c"],"text":"A027465 \u2192 Veronese 2 (j=1)"},"headline":"A027465 \u2192 Veronese 2 (j=1)","chain":null}}],"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:     9;\n  4:    54,     2;\n  5:   270,    30;\n  6:  1215,   270,    3;\n  7:  5103,  1890,   63;\n  8: 20412, 11340,  756,   4;\n  9: 78732, 61236, 6804, 108;\n","recurrence_sentence":"The row polynomials satisfy (3*n^2-9*n+6)*P(n,x) = (9*n^2-9*n+(n^2-3*n)*x)*P(n-1,x) + (18*n*x-2*n*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 0;\nP[2, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((9*n^2 - 9*n + (n^2 - 3*n)*x)*P[n - 1, x] + (18*n*x - 2*n*x^2)*D[P[n - 1, x], x])/(3*n^2 - 9*n + 6)]];\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; for(n = 3, N, my(i = n + 1); v[i] = ((9*n^2 - 9*n + (n^2 - 3*n)*x)*v[i - 1] + (18*n*x - 2*n*x^2)*deriv(v[i - 1], x))/(3*n^2 - 9*n + 6)); 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","D"],"edges":["e-5d982f8f22366510","e-7375e7ec9839fed4"],"text":"A027465 \u2192 Veronese 2 \u2192 D"},"headline":"A027465 \u2192 Veronese 2 \u2192 D","chain":null}
