{"id":"n-288ee4e6a60f9c49","revision":1,"first_index":1,"fingerprint":"f6-08790148ebd3c793","name":"obreschkoff_combination of Galton triangle T(n, k) = T(n-1, k-1) + (3k-2)*T(n-1, k) read by rows.","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","1"],["2","6","1"],["2","26","13","1"],["2","106","117","23","1"],["2","426","925","347","36","1"],["2","1706","6901","4395","815","52","1"],["2","6826","50013","50851","14990","1647","71","1"],["2","27306","356917","558523","245721","41342","2996","93","1"],["2","109226","2525725","5942147","3752896","907193","98266","5042","118","1"],["2","436906","17789301","61947195","54729795","18267984","2774247","209190","7992","146","1"],["2","1747626","124962013","637261251","773434530","347017539","70978677","7376427","408990","12080","177","1"],["2","6990506","876481717","6497574523","10691910141","6325715154","1695612402","233260071","17601177","747230","17567","211","1"],["2","27962026","6142362525","65852226947","145492406356","111903352605","38542350792","6827333964","673289496","38523617","1291807","24741","248","1"],["2","111848106","43024499701","664664631995","1957253509575","1935946048036","844208017653","188743698000","23659571364","1751950772","78569634","2133001","33917","288","1"],["2","447392426","301283346013","6689670819651","26108960256470","32932390278151","17975898383443","4996569373653","780232982100","72714192980","4187609426","151091668","3387930","45437","331","1"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["3","9","42","249","1737","13872","124401","1232901","13340616","156162759","1963199313","26345910702","375501125127","5659444512045","89862509159682"],"nonn":true,"definitions":[{"position":0,"source":"derived:obreschkoff_combination","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["2","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","3"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162755,"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-45f39b72ccd15e60"]}],"region_exact":{},"incoming_edges":[{"id":"e-45f39b72ccd15e60","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-288ee4e6a60f9c49","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":"8f5a26bc17f09b82bb19037d3b9ba7689096abce219958a3896d8621cd0611cf","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{}}],"premises":[{"position":0,"node_id":"n-cab14672969f6507","role":"sequence","name":"Galton triangle T(n, k) = T(n-1, k-1) + (3k-2)*T(n-1, k) 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-cab14672969f6507","role":"sequence","name":"Galton triangle T(n, k) = T(n-1, k-1) + (3k-2)*T(n-1, k) 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,      1;\n   2: 2,      6,        1;\n   3: 2,     26,       13,        1;\n   4: 2,    106,      117,       23,        1;\n   5: 2,    426,      925,      347,       36,        1;\n   6: 2,   1706,     6901,     4395,      815,       52,       1;\n   7: 2,   6826,    50013,    50851,    14990,     1647,      71,      1;\n   8: 2,  27306,   356917,   558523,   245721,    41342,    2996,     93,    1;\n   9: 2, 109226,  2525725,  5942147,  3752896,   907193,   98266,   5042,  118,   1;\n  10: 2, 436906, 17789301, 61947195, 54729795, 18267984, 2774247, 209190, 7992, 146, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+x)*P(n-1,x) + 3*x*d/dx P(n-1,x) for n >= 2, with P(1,x) = 2 + x.","mathematica":"P[1, x_] := 2 + x;\nP[n_, x_] := P[n, x] = Expand[(1 + x)*P[n - 1, x] + 3*x*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 + x; for(n = 2, N, my(i = n); v[i] = (1 + x)*v[i - 1] + 3*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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-cab14672969f6507","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-45f39b72ccd15e60"],"text":"A111577 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A111577 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}