{"id":"n-54cdd4c005ce1726","revision":0,"first_index":0,"fingerprint":"f6-0e61d18a7a9911bc","name":"shift_x_plus_one of reversal of Coefficient triangle of certain polynomials N(4; m,x).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["6","1"],["28","14","1"],["120","108","24","1"],["495","660","270","36","1"],["2002","3575","2200","550","50","1"],["8008","18018","15015","5720","990","66","1"],["31824","86632","91728","47775","12740","1638","84","1"],["125970","403104","519792","346528","127400","25480","2548","104","1"],["497420","1831410","2790720","2279088","1079568","299880","47040","3780","126","1"],["1961256","8171900","14389650","13953600","8139600","2930256","642600","81600","5400","150","1"],["7726160","35956360","71912720","80901810","56279520","25069968","7162848","1279080","134640","7480","176","1"],["30421755","156454740","350574510","449454500","364058145","194164344","68942412","16116408","2398275","213180","10098","204","1"],["119759850","674647155","1674986040","2412777510","2234053250","1391987025","593914464","173225052","33891858","4279275","326040","13338","234","1"],["471435600","2887543050","7870883475","12593413560","13136233110","9383023650","4691511825","1649762400","404191788","67365298","7322315","484120","17290","266","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","7","43","253","1462","8378","47818","272422","1550927","8829033","50276013","286430763","1632808572","9313861092","53163187748"],"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":[["0"],["4"],["1"]]},"terms":[{"coeff":{"coeffs":[["12","-4"],["14","6"],["4","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-4","-4"],["0","-2","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":176745,"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-e8e9405136db9065"]},{"id":176746,"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-e8e9405136db9065"]},{"id":176747,"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-e8e9405136db9065"]},{"id":176748,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-e8e9405136db9065"]}],"region_exact":{},"incoming_edges":[{"id":"e-e8e9405136db9065","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-54cdd4c005ce1726","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":"7ddf820344a3f08491b0498419f922be92403c8f6b4a0f892edf57b1c28d3d18","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-ef179a3c7df58b5a","role":"sequence","name":"reversal of Coefficient triangle of certain polynomials N(4; m,x).","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-3704ca4400e535a5","kind":"fingerprint_related","node_a_id":"n-98f8cf8c446dcc87","node_b_id":"n-54cdd4c005ce1726","fingerprint":"f6-0e61d18a7a9911bc","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-0e61d18a7a9911bc"},"other_id":"n-98f8cf8c446dcc87","other":{"id":"n-98f8cf8c446dcc87","name":"mobius_x_over_x_plus_one of Coefficient triangle of certain polynomials N(4; m,x).","first_index":0,"fingerprint":"f6-0e61d18a7a9911bc","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","6"],["1","14","28"]],"card_rows":[["1"],["1","6"],["1","14","28"],["1","24","108","120"]],"row_sums_preview":["1","7","43","253","1462","8378","47818","272422"],"regions":["[-1, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-9516b04042fe64c0","steps":["x \u21a6 x/(x+1)"],"edges":["e-c45bb838d2ec9765"],"text":"A062264 \u2192 x \u21a6 x/(x+1)"},"headline":"A062264 \u2192 x \u21a6 x/(x+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:      1;\n  1:      6,       1;\n  2:     28,      14,       1;\n  3:    120,     108,      24,       1;\n  4:    495,     660,     270,      36,       1;\n  5:   2002,    3575,    2200,     550,      50,      1;\n  6:   8008,   18018,   15015,    5720,     990,     66,     1;\n  7:  31824,   86632,   91728,   47775,   12740,   1638,    84,    1;\n  8: 125970,  403104,  519792,  346528,  127400,  25480,  2548,  104,   1;\n  9: 497420, 1831410, 2790720, 2279088, 1079568, 299880, 47040, 3780, 126, 1;\n","recurrence_sentence":"The row polynomials satisfy (n^2+4*n)*P(n,x) = (4*n^2+14*n+12+(3*n^2+6*n-4)*x)*P(n-1,x) - ((2*n+4)*x+(2*n+4)*x^2)*d/dx 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[((4*n^2 + 14*n + 12 + (3*n^2 + 6*n - 4)*x)*P[n - 1, x] - ((2*n + 4)*x + (2*n + 4)*x^2)*D[P[n - 1, x], x])/(n^2 + 4*n)]];\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] = ((4*n^2 + 14*n + 12 + (3*n^2 + 6*n - 4)*x)*v[i - 1] - ((2*n + 4)*x + (2*n + 4)*x^2)*deriv(v[i - 1], x))/(n^2 + 4*n)); 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-9516b04042fe64c0","steps":["reverse","x \u21a6 x+1"],"edges":["e-6d0521392d648070","e-e8e9405136db9065"],"text":"A062264 \u2192 reverse \u2192 x \u21a6 x+1"},"headline":"A062264 \u2192 reverse \u2192 x \u21a6 x+1","chain":null}
