{"id":"n-c6da825af22846f3","revision":0,"first_index":0,"fingerprint":"f6-3e306556aac30f2d","name":"shift_x_plus_one of Triangle of coefficients of polynomials v(n,x) jointly generated with A208508; see the Formula section.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["3"],["6","1"],["12","5"],["24","16","1"],["48","44","7"],["96","112","30","1"],["192","272","104","9"],["384","640","320","48","1"],["768","1472","912","200","11"],["1536","3328","2464","720","70","1"],["3072","7424","6400","2352","340","13"],["6144","16384","16128","7168","1400","96","1"],["12288","35840","39680","20736","5152","532","15"],["24576","77824","95744","57600","17472","2464","126","1"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","3","7","17","41","99","239","577","1393","3363","8119","19601","47321","114243","275807"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["1"],["2"],["1"]]},"terms":[{"coeff":{"coeffs":[["16","-1"],["-9","0"],["5","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["24","40","-2"],["-12","-14","-2"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","12","12"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":172284,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-330b31b4167e8f5f"]},{"id":172285,"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-330b31b4167e8f5f"]},{"id":172286,"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-330b31b4167e8f5f"]}],"region_exact":{},"incoming_edges":[{"id":"e-330b31b4167e8f5f","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-c6da825af22846f3","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"75ef707f167a9743ca20e2ef632841537fc4f180c0b2a9cc05503ba6469a5ef6","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}}],"premises":[{"position":0,"node_id":"n-d4c5acf24bc40277","role":"sequence","name":"Triangle of coefficients of polynomials v(n,x) jointly generated with A208508; see the Formula section.","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-13284bdc155bf83d","kind":"candidate_equal","node_a_id":"n-c6da825af22846f3","node_b_id":"n-74cd725b7143f347","fingerprint":null,"oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"reason":"no all-index equality certificate was found","rows_compared":15},"other_id":"n-74cd725b7143f347","other":{"id":"n-74cd725b7143f347","name":"basis_map_A007318 of Triangle of coefficients of polynomials v(n,x) jointly generated with A208508; see the Formula section.","first_index":0,"fingerprint":"f6-3e306556aac30f2d","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3"],["6","1"]],"card_rows":[["1"],["3"],["6","1"],["12","5"]],"row_sums_preview":["1","3","7","17","41","99","239","577"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d4c5acf24bc40277","steps":["basis A007318"],"edges":["e-87aabae02c8b38a2"],"text":"A208509 \u2192 basis A007318"},"headline":"A208509 \u2192 basis A007318","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:   1;\n  1:   3;\n  2:   6,    1;\n  3:  12,    5;\n  4:  24,   16,   1;\n  5:  48,   44,   7;\n  6:  96,  112,  30,   1;\n  7: 192,  272, 104,   9;\n  8: 384,  640, 320,  48,  1;\n  9: 768, 1472, 912, 200, 11;\n","recurrence_sentence":"The row polynomials satisfy (n^2+2*n+1)*P(n,x) = (5*n^2-9*n+16+(n^2-1)*x)*P(n-1,x) - (12*n-24+(14*n-40)*x+(2*n+2)*x^2)*d/dx P(n-1,x) + (12*x+12*x^2)*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((5*n^2 - 9*n + 16 + (n^2 - 1)*x)*P[n - 1, x] - (12*n - 24 + (14*n - 40)*x + (2*n + 2)*x^2)*D[P[n - 1, x], x] + (12*x + 12*x^2)*D[P[n - 1, x], {x, 2}])/(n^2 + 2*n + 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((5*n^2 - 9*n + 16 + (n^2 - 1)*x)*v[i - 1] - (12*n - 24 + (14*n - 40)*x + (2*n + 2)*x^2)*deriv(v[i - 1], x) + (12*x + 12*x^2)*deriv(deriv(v[i - 1], x), x))/(n^2 + 2*n + 1)); 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-d4c5acf24bc40277","steps":["x \u21a6 x+1"],"edges":["e-330b31b4167e8f5f"],"text":"A208509 \u2192 x \u21a6 x+1"},"headline":"A208509 \u2192 x \u21a6 x+1","chain":null}
