{"id":"n-e5ed5ba4147e82ac","revision":1,"first_index":0,"fingerprint":"f6-5ca6547852b7bd7f","name":"obreschkoff_combination of Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) =  4*x + 2.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["3","2"],["6","26","4"],["12","208","188","8"],["24","1400","3864","1224","16"],["48","8800","57504","55040","7600","32"],["96","53792","728576","1560896","682336","46240","64"],["192","325120","8438848","34751616","33919296","7810304","278976","128"],["384","1956224","92656768","669478272","1230048128","643997824","85171584","1677440","256"],["768","11749888","984138752","11750715392","36736410112","36175840256","11199374336","900923392","10072832","512"],["1536","70527488","10234174464","193694607360","964229047296","1599282293760","939803922432","183479371776","9345000960","60455424","1024"],["3072","423227392","104981498880","3056089450496","23122503145472","60112007266304","59476954019840","22376393830400","2882844253184","95707025408","362773504","2048"],["6144","2539501568","1067348719616","46737209550848","519381777432576","2012424496459776","3103058022690816","1976070045585408","500061473052672","43959346796544","972064921600","2176731136","4096"],["12288","15237308416","10788853178368","698766993653760","11113391919419392","61843544100003840","140818050425438208","139727416899207168","60386855831851008","10660067655917568","655874995576832","9819327758336","13060583424","8192"],["24576","91424497664","108641690288128","10275138348761088","229213813482102784","1780878111283109888","5756134305534001152","8404185980706914304","5678250432734797824","1731604647370678272","219252512819838976","9630052206362624","98837599838208","78363926528","16384"],["49152","548548378624","1091301583437824","149244752792944640","4595885393547149312","48763833798653968384","217093301501169713152","447283051646694162432","444634999971521249280","213195929179718025216","47261442964765392896","4386908019076071424","139723342276870144","992555408097280","470184476672","32768"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["5","36","416","6528","129024","3072000","85524480","2724986880","97769226240","3900199403520","171228266496000","8203736501452800","425924605299916800","23819632643565158400","1427506405445940019200"],"nonn":true,"definitions":[{"position":0,"source":"derived:obreschkoff_combination","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["3","2"],["6","26","4"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","-6"],["0","4"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","4","-4"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1","2"],["0","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","4","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162849,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, +inf)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)","region_exact":null,"edges":["e-697623ff3ff432a2"]}],"region_exact":{},"incoming_edges":[{"id":"e-697623ff3ff432a2","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-e5ed5ba4147e82ac","premise_count":2,"theorem":"Hermite--Kakeya--Obreschkoff: real-rooted f, g with interlacing zeros give a zero or real-rooted a f + b g for all real a, b (Obreschkoff 1963; Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"96581031b41046bd185d9953e4c9339d2b944bc9b3dd5ca9ad18e17c08448ebc","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-2c43e713cd32b015","role":"sequence","name":"Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) =  4*x + 2.","uses":{"claim":"real_rooted","from_index":0,"region":{}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)"},{"position":1,"node_id":"n-2c43e713cd32b015","role":"sequence","name":"Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) =  4*x + 2.","uses":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"uses_text":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1"}]}],"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:    3,        2;\n  1:    6,       26,           4;\n  2:   12,      208,         188,            8;\n  3:   24,     1400,        3864,         1224,           16;\n  4:   48,     8800,       57504,        55040,         7600,            32;\n  5:   96,    53792,      728576,      1560896,       682336,         46240,           64;\n  6:  192,   325120,     8438848,     34751616,     33919296,       7810304,       278976,          128;\n  7:  384,  1956224,    92656768,    669478272,   1230048128,     643997824,     85171584,      1677440,        256;\n  8:  768, 11749888,   984138752,  11750715392,  36736410112,   36175840256,  11199374336,    900923392,   10072832,      512;\n  9: 1536, 70527488, 10234174464, 193694607360, 964229047296, 1599282293760, 939803922432, 183479371776, 9345000960, 60455424, 1024;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2+(4*n-6)*x)*P(n-2,x) + (4*x-4*x^2)*d/dx P(n-2,x) + (1+(4*n+2)*x)*P(n-1,x) + (4*x-4*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 3 + 2*x, P(1,x) = 6 + 26*x + 4*x^2.","mathematica":"P[0, x_] := 3 + 2*x;\nP[1, x_] := 6 + 26*x + 4*x^2;\nP[n_, x_] := P[n, x] = Expand[(2 + (4*n - 6)*x)*P[n - 2, x] + (4*x - 4*x^2)*D[P[n - 2, x], x] + (1 + (4*n + 2)*x)*P[n - 1, x] + (4*x - 4*x^2)*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] = 3 + 2*x; v[2] = 6 + 26*x + 4*x^2; for(n = 2, N, my(i = n + 1); v[i] = (2 + (4*n - 6)*x)*v[i - 2] + (4*x - 4*x^2)*deriv(v[i - 2], x) + (1 + (4*n + 2)*x)*v[i - 1] + (4*x - 4*x^2)*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-2c43e713cd32b015","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-697623ff3ff432a2"],"text":"A257612 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A257612 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}