{"id":"n-db68edc2329f4b20","revision":1,"first_index":1,"fingerprint":"f6-68a37c0728e7acf6","name":"obreschkoff_combination of Triangle T(n, k) = coefficients of ( t(n, x) ) where t(n, x) = (1-x)^(n+1)*p(n, x)/x, p(n, x) = x*D( p(n-1, x) ), with p(1, x) = x/(1-x)^2, p(2, x) = x*(1+x)/(1-x)^3, and p(3, x) = x*(1+6*x+x^2)/(1-x)^4, read by rows.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["2","21","16","1"],["2","49","105","35","1"],["2","107","496","440","74","1"],["2","225","1989","3654","1656","153","1"],["2","463","7244","24155","22490","5813","312","1"],["2","941","24821","138501","229977","123255","19471","631","1"],["2","1899","81680","722090","1959984","1868508","623656","63150","1270","1"],["2","3817","261501","3522640","14736956","22761882","13590538","2980664","200250","2549","1"],["2","7655","821404","16381549","101261404","237979070","229954144","91193506","13662386","624971","5108","1"],["2","15333","2545869","73541255","650822817","2224979610","3254521530","2096286870","576010860","60700273","1928553","10227","1"],["2","30691","7816496","321547156","3976063230","19119007651","40364519488","39334977568","17673543598","3473593625","263392752","5903276","20466","1"],["2","61409","23838245","1378063130","23357257016","153912130751","452614693767","634558423020","432398723850","140214650383","20205210547","1122486874","17965660","40945","1"],["2","122847","72353996","5816256655","133061573334","1176989095907","4690313167736","9112672662003","8930728932750","4411826574637","1060131702540","114234425917","4717608026","54449745","81904","1"],["2","245725","218863749","24260037369","739802609361","8638729714209","45642271094521","119389225245677","161793382895817","115148809480663","42407433615791","7711659585003","631384440099","19614783339","164536851","163823","1"],["2","491483","660440944","100268696910","4033953907140","61336694374854","422120466270856","1452903758565118","2641517257314080","2599113099663940","1383397220178976","388351762369274","54364837399212","3426829504410","80872777320","496149818","327662","1"]],"degrees":[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],"row_sums":["40","192","1120","7680","60480","537600","5322240","58060800","691891200","8941363200","124540416000","1859803545600","29640619008000","502146957312000","9010748178432000"],"nonn":true,"definitions":[{"position":0,"source":"derived:obreschkoff_combination","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["2","21","16","1"],["2","49","105","35","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","0"],["0","1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","1","-1"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","2"],["0","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","-1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162806,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, +inf)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)","region_exact":null,"edges":["e-83898f0435e2f8a0"]}],"region_exact":{},"incoming_edges":[{"id":"e-83898f0435e2f8a0","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-db68edc2329f4b20","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":"57657a95332f4b0251e0df86751098ac08fa1834d624f4097ca3c5e99794371a","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{}}],"premises":[{"position":0,"node_id":"n-9a3eef8528def021","role":"sequence","name":"Triangle T(n, k) = coefficients of ( t(n, x) ) where t(n, x) = (1-x)^(n+1)*p(n, x)/x, p(n, x) = x*D( p(n-1, x) ), with p(1, x) = x/(1-x)^2, p(2, x) = x*(1+x)/(1-x)^3, and p(3, x) = x*(1+6*x+x^2)/(1-x)^4, read by rows.","uses":{"claim":"real_rooted","from_index":1,"region":{}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)"},{"position":1,"node_id":"n-9a3eef8528def021","role":"sequence","name":"Triangle T(n, k) = coefficients of ( t(n, x) ) where t(n, x) = (1-x)^(n+1)*p(n, x)/x, p(n, x) = x*D( p(n-1, x) ), with p(1, x) = x/(1-x)^2, p(2, x) = x*(1+x)/(1-x)^3, and p(3, x) = x*(1+6*x+x^2)/(1-x)^4, read by rows.","uses":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"uses_text":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 2,    21,      16,        1;\n   2: 2,    49,     105,       35,         1;\n   3: 2,   107,     496,      440,        74,          1;\n   4: 2,   225,    1989,     3654,      1656,        153,          1;\n   5: 2,   463,    7244,    24155,     22490,       5813,        312,          1;\n   6: 2,   941,   24821,   138501,    229977,     123255,      19471,        631,         1;\n   7: 2,  1899,   81680,   722090,   1959984,    1868508,     623656,      63150,      1270,        1;\n   8: 2,  3817,  261501,  3522640,  14736956,   22761882,   13590538,    2980664,    200250,     2549,       1;\n   9: 2,  7655,  821404, 16381549, 101261404,  237979070,  229954144,   91193506,  13662386,   624971,    5108,     1;\n  10: 2, 15333, 2545869, 73541255, 650822817, 2224979610, 3254521530, 2096286870, 576010860, 60700273, 1928553, 10227, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+n*x)*P(n-2,x) + (x-x^2)*d/dx P(n-2,x) + (n+2)*x*P(n-1,x) + (x-x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 2 + 21*x + 16*x^2 + x^3, P(2,x) = 2 + 49*x + 105*x^2 + 35*x^3 + x^4.","mathematica":"P[1, x_] := 2 + 21*x + 16*x^2 + x^3;\nP[2, x_] := 2 + 49*x + 105*x^2 + 35*x^3 + x^4;\nP[n_, x_] := P[n, x] = Expand[(1 + n*x)*P[n - 2, x] + (x - x^2)*D[P[n - 2, x], x] + (n + 2)*x*P[n - 1, x] + (x - x^2)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 2 + 21*x + 16*x^2 + x^3; v[2] = 2 + 49*x + 105*x^2 + 35*x^3 + x^4; for(n = 3, N, my(i = n); v[i] = (1 + n*x)*v[i - 2] + (x - x^2)*deriv(v[i - 2], x) + (n + 2)*x*v[i - 1] + (x - 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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-9a3eef8528def021","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-83898f0435e2f8a0"],"text":"A166344 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A166344 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}