{"id":"n-3dd3bd37c05e36d2","revision":0,"first_index":0,"fingerprint":"f6-429c048c3a049f47","name":"shift_x_plus_one of reversal of Triangle of coefficients of polynomials v(n,x) jointly generated with A207606; see Formula section.","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"],["3","1"],["8","7","1"],["21","30","12","1"],["55","109","73","18","1"],["144","365","344","145","25","1"],["377","1164","1416","850","255","33","1"],["987","3593","5358","4184","1806","413","42","1"],["2584","10835","19139","18422","10415","3458","630","52","1"],["6765","32106","65556","75051","52557","22995","6132","918","63","1"],["17711","93845","217487","288730","241375","131389","46425","10248","1290","75","1"],["46368","271321","703618","1062895","1033370","673757","297374","87417","16335","1760","88","1"],["121393","777432","2230608","3778372","4189578","3185142","1689058","622512","155628","25047","2343","102","1"],["317811","2211025","6954060","13054880","16262700","14131190","8759200","3889730","1223365","264550","37180","3055","117","1"],["832040","6248479","21377365","44055740","60920930","59576300","42291060","22001820","8359845","2281565","432575","53690","3913","133","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","16","64","256","1024","4096","16384","65536","262144","1048576","4194304","16777216","67108864","268435456"],"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"],["3","1"]],"recurrence":{"denominator":{"coeffs":[["8"],["-16"],["8"]]},"terms":[{"coeff":{"coeffs":[["56","66","18"],["-100","-114","-30"],["44","48","12"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["68","174","148","42"],["-50","-120","-94","-24"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["15","57","81","51","12"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":172960,"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-8353d9cef1eb34a3"]},{"id":172961,"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-8353d9cef1eb34a3"]},{"id":172962,"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-8353d9cef1eb34a3"]}],"region_exact":{},"incoming_edges":[{"id":"e-8353d9cef1eb34a3","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-3dd3bd37c05e36d2","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":"fc8d843d5e45885bb8ca0af535362fa6113f4cdc919b7afc72c4b4c22f717e24","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-e860d500a070a4fd","role":"sequence","name":"reversal of Triangle of coefficients of polynomials v(n,x) jointly generated with A207606; see 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-0d9552042c10645b","kind":"fingerprint_related","node_a_id":"n-106df0aaa5357879","node_b_id":"n-3dd3bd37c05e36d2","fingerprint":"f6-429c048c3a049f47","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-429c048c3a049f47"},"other_id":"n-106df0aaa5357879","other":{"id":"n-106df0aaa5357879","name":"mobius_x_over_x_plus_one of Triangle of coefficients of polynomials v(n,x) jointly generated with A207606; see Formula section.","first_index":0,"fingerprint":"f6-429c048c3a049f47","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","3"],["1","7","8"]],"card_rows":[["1"],["1","3"],["1","7","8"],["1","12","30","21"]],"row_sums_preview":["1","4","16","64","256","1024","4096","16384"],"regions":["[-1, 0)"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-353fb5a029e4e669","steps":["x \u21a6 x/(x+1)"],"edges":["e-ba8938d63a27da5b"],"text":"A207607 \u2192 x \u21a6 x/(x+1)"},"headline":"A207607 \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:    3,     1;\n  2:    8,     7,     1;\n  3:   21,    30,    12,     1;\n  4:   55,   109,    73,    18,     1;\n  5:  144,   365,   344,   145,    25,     1;\n  6:  377,  1164,  1416,   850,   255,    33,    1;\n  7:  987,  3593,  5358,  4184,  1806,   413,   42,   1;\n  8: 2584, 10835, 19139, 18422, 10415,  3458,  630,  52,  1;\n  9: 6765, 32106, 65556, 75051, 52557, 22995, 6132, 918, 63, 1;\n","recurrence_sentence":"The row polynomials satisfy (8*n^2-16*n+8)*P(n,x) = (44*n^2-100*n+56+(48*n^2-114*n+66)*x+(12*n^2-30*n+18)*x^2)*P(n-1,x) - (50*n-68+(120*n-174)*x+(94*n-148)*x^2+(24*n-42)*x^3)*d/dx P(n-1,x) + (15+57*x+81*x^2+51*x^3+12*x^4)*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + x;\nP[n_, x_] := P[n, x] = Expand[Together[((44*n^2 - 100*n + 56 + (48*n^2 - 114*n + 66)*x + (12*n^2 - 30*n + 18)*x^2)*P[n - 1, x] - (50*n - 68 + (120*n - 174)*x + (94*n - 148)*x^2 + (24*n - 42)*x^3)*D[P[n - 1, x], x] + (15 + 57*x + 81*x^2 + 51*x^3 + 12*x^4)*D[P[n - 1, x], {x, 2}])/(8*n^2 - 16*n + 8)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + x; for(n = 2, N, my(i = n + 1); v[i] = ((44*n^2 - 100*n + 56 + (48*n^2 - 114*n + 66)*x + (12*n^2 - 30*n + 18)*x^2)*v[i - 1] - (50*n - 68 + (120*n - 174)*x + (94*n - 148)*x^2 + (24*n - 42)*x^3)*deriv(v[i - 1], x) + (15 + 57*x + 81*x^2 + 51*x^3 + 12*x^4)*deriv(deriv(v[i - 1], x), x))/(8*n^2 - 16*n + 8)); 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-353fb5a029e4e669","steps":["reverse","x \u21a6 x+1"],"edges":["e-a355ebab7822b2ca","e-8353d9cef1eb34a3"],"text":"A207607 \u2192 reverse \u2192 x \u21a6 x+1"},"headline":"A207607 \u2192 reverse \u2192 x \u21a6 x+1","chain":null}