{"id":"n-1bf471426fab0fa9","revision":1,"first_index":0,"fingerprint":"f6-6319fdf993d134a2","name":"shift_x_plus_one of Triangular array read by rows. T(n,k) = A008277(n,k)*2^k; n >= 1, 1 <= k <= n.","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":[["2"],["6","4"],["22","28","8"],["94","172","96","16"],["454","1068","872","288","32"],["2430","6924","7360","3600","800","64"],["14214","47276","61448","39520","13120","2112","128"],["89918","340428","520352","412496","181440","43904","5376","256"],["610182","2582252","4520104","4241440","2314592","746368","137984","13312","512"],["4412798","20589580","40489344","43747216","28416864","11428032","2827776","413184","32256","1024"],["33827974","172162604","374867528","457129440","343552640","165224640","51198336","10045440","1190400","76800","2048"],["273646526","1506041420","3590051168","4866722256","4145250560","2316092416","861552384","212374272","33876480","3322880","180224","4096"],["2326980998","13751561068","35562505448","52961438112","50330660512","31988554752","13872776192","4117859328","827295744","109429760","9033728","417792","8192"],["20732504062","130785217164","364199963712","590216281616","618180273568","439806416192","217657110528","75542511616","18320839680","3058085888","340893696","24014848","958464","16384"],["192982729350","1293005804204","3853219097736","6742418711392","7706726559168","6060754539840","3367324408448","1337306054656","380137032704","76747646976","10811953152","1029496832","62619648","2179072","32768"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["2","10","58","378","2714","21178","177818","1594170","15166746","152358074","1609277850","17809114682","205858521626","2478865087418","31022528865946"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["2"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["3","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["1","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":165908,"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-c012cb3d271aa10d"]},{"id":165909,"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-c012cb3d271aa10d"]},{"id":165910,"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-c012cb3d271aa10d"]},{"id":165911,"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-c012cb3d271aa10d"]}],"region_exact":{},"incoming_edges":[{"id":"e-c012cb3d271aa10d","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-1bf471426fab0fa9","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":"19b02450de551cc601b0ea0e297ae86d639630843ab091d9998e2e0d8bcf8031","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-d5fb64ec7bd8ef70","role":"sequence","name":"Triangular array read by rows. T(n,k) = A008277(n,k)*2^k; n >= 1, 1 <= k <= n.","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:       2;\n  1:       6,        4;\n  2:      22,       28,        8;\n  3:      94,      172,       96,       16;\n  4:     454,     1068,      872,      288,       32;\n  5:    2430,     6924,     7360,     3600,      800,       64;\n  6:   14214,    47276,    61448,    39520,    13120,     2112,     128;\n  7:   89918,   340428,   520352,   412496,   181440,    43904,    5376,    256;\n  8:  610182,  2582252,  4520104,  4241440,  2314592,   746368,  137984,  13312,   512;\n  9: 4412798, 20589580, 40489344, 43747216, 28416864, 11428032, 2827776, 413184, 32256, 1024;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3+2*x)*P(n-1,x) + (1+x)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 2.","mathematica":"P[0, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[(3 + 2*x)*P[n - 1, x] + (1 + x)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 2; for(n = 1, N, my(i = n + 1); v[i] = (3 + 2*x)*v[i - 1] + (1 + x)*deriv(v[i - 1], x)); 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-d5fb64ec7bd8ef70","steps":["x \u21a6 x+1"],"edges":["e-c012cb3d271aa10d"],"text":"A227450 \u2192 x \u21a6 x+1"},"headline":"A227450 \u2192 x \u21a6 x+1","chain":null}
