{"id":"n-4484b873e17cc5c5","revision":1,"first_index":0,"fingerprint":"f6-e61db7f606ab2bcb","name":"veronese_section_2_1 of Triangular array read by rows: row n gives the coefficients of the polynomial p(n,x) defined in Comments.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"irregular","degree_slope":null,"degree_intercept":null,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[[],["1"],["2"],["7"],["16"],["41","1"],["94","4"],["219","18"],["492","60"],["1101","195","1"],["2426","576","6"],["5311","1644","33"],["11528","4488","140"],["24881","11925","546","1"],["53398","30860","1932","8"]],"degrees":[-1,0,0,0,0,1,1,1,1,2,2,2,2,3,3],"row_sums":["0","1","2","7","16","42","98","237","552","1297","3008","6988","16156","37353","86198"],"nonn":true,"definitions":[{"position":0,"source":"derived:veronese_section_2_1","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["1"],["2"],["7"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["3","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-4","0"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["-4","1"]]},"deriv_order":0,"offset":4,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":163175,"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-d9167ed20c29eacc"]},{"id":163176,"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-d9167ed20c29eacc"]},{"id":163177,"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-d9167ed20c29eacc"]}],"region_exact":{},"incoming_edges":[{"id":"e-d9167ed20c29eacc","operation":{"index_shift":0,"modulus":2,"operation":"veronese_section","residue":1},"operation_detail":"r=2, j=1, shift 0","conclusion_id":"n-4484b873e17cc5c5","premise_count":1,"theorem":"Veronese sections of Polya frequency sequences (Aissen--Schoenberg--Whitney--Edrei): if P is real-rooted with coefficients of one sign, equivalently nonzero with all roots in (-inf, 0], every section V_(r,j)(P) is zero or real-rooted with roots in (-inf, 0]","lean_route":null,"lean_declarations":[],"certificate_hash":"be9a09db7c09dc06b797d75d404b8c5a5f1b30dc201541c421a61aa074e246a0","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-ac04e52510a48bb5","role":"sequence","name":"Triangular array read by rows: row n gives the coefficients of the polynomial p(n,x) defined in Comments.","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:    2;\n  3:    7;\n  4:   16;\n  5:   41,   1;\n  6:   94,   4;\n  7:  219,  18;\n  8:  492,  60;\n  9: 1101, 195, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = 2*P(n-1,x) + 3*P(n-2,x) - 4*P(n-3,x) - (4-x)*P(n-4,x) for n >= 4, with P(0,x) = 0, P(1,x) = 1, P(2,x) = 2, P(3,x) = 7.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[2, x_] := 2;\nP[3, x_] := 7;\nP[n_, x_] := P[n, x] = Expand[2*P[n - 1, x] + 3*P[n - 2, x] - 4*P[n - 3, x] - (4 - x)*P[n - 4, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; v[3] = 2; v[4] = 7; for(n = 4, N, my(i = n + 1); v[i] = 2*v[i - 1] + 3*v[i - 2] - 4*v[i - 3] - (4 - x)*v[i - 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-75ccd0453a9b6cec","steps":["x \u21a6 x+2 (shift +1)","Veronese 2 (j=1)"],"edges":["e-7fc1b55fb0aa373a","e-d9167ed20c29eacc"],"text":"A011973 \u2192 x \u21a6 x+2 (shift +1) \u2192 Veronese 2 (j=1)"},"headline":"A011973 \u2192 x \u21a6 x+2 (shift +1) \u2192 Veronese 2 (j=1)","chain":null}