{"id":"n-4ca244aa56ebfc37","revision":1,"first_index":0,"fingerprint":"f6-b5b6960cbd68b6b0","name":"obreschkoff_combination of Triangle read by rows: T(n, k) is the Sheffer triangle ((1 - 4*x)^(-1/4), (-1/4)*log(1 - 4*x)). A generalized Stirling1 triangle.","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":true,"rows":[["2","1"],["6","7","1"],["50","65","16","1"],["630","871","269","29","1"],["10530","15201","5384","758","46","1"],["218790","326503","127249","21190","1720","67","1"],["5429970","8323809","3487208","654079","64010","3391","92","1"],["156633750","245571975","108964365","22373659","2503669","162085","6055","121","1"],["5148029250","8228462145","3827935664","844760636","104745396","7839174","361536","10044","154","1"],["189871431750","308650887495","149439684609","34997119036","4710574952","394159794","21192150","732684","15738","191","1"],["7764742235250","12812594211585","6420788610912","1581029482701","227721475820","20840404850","1261615656","51192498","1377330","23565","232","1"],["348673901325750","583128841814055","301165212309741","77430418938697","11810062811041","1163993776550","77529763058","3562387026","113110008","2436995","34001","277","1"],["17054701695281250","28871939413310625","15315123959940600","4089068318768850","655231043859986","68763942160111","4957989050080","251884848380","9099330438","232429263","4102120","47570","326","1"],["902534813714283750","1544985022970391975","839393948725974225","231732210600471906","38769965844923664","4295145818941549","331232205878431","18293838345068","733702368542","21408426897","449706543","6622226","64844","379","1"],["51377629951068671250","88853475833003736225","49330359165075348600","14032077213932015211","2439045687384105254","283322795915355833","23155854753747716","1372987499822939","60078923023262","1953062639099","47025426448","826984249","10317034","86443","436","1"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["3","14","132","1800","31920","695520","17962560","536215680","18162144000","688085798400","28829030630400","1323453638707200","66060047583129600","3562061389286400000","206340501568481280000"],"nonn":true,"definitions":[{"position":0,"source":"derived:obreschkoff_combination","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["2","1"]],"recurrence":{"denominator":{"coeffs":[["-2","1"],["4","0"]]},"terms":[{"coeff":{"coeffs":[["-6","-1","1"],["-4","8","0"],["16","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162830,"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-0f7f0ded98411894"]}],"region_exact":{},"incoming_edges":[{"id":"e-0f7f0ded98411894","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-4ca244aa56ebfc37","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":"9864f2e57c14e55521f44898ec130fb17d1a37cc08178577cced1549f904fa28","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-4172472abe953cea","role":"sequence","name":"Triangle read by rows: T(n, k) is the Sheffer triangle ((1 - 4*x)^(-1/4), (-1/4)*log(1 - 4*x)). A generalized Stirling1 triangle.","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-4172472abe953cea","role":"sequence","name":"Triangle read by rows: T(n, k) is the Sheffer triangle ((1 - 4*x)^(-1/4), (-1/4)*log(1 - 4*x)). A generalized Stirling1 triangle.","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:            2,            1;\n  1:            6,            7,            1;\n  2:           50,           65,           16,           1;\n  3:          630,          871,          269,          29,          1;\n  4:        10530,        15201,         5384,         758,         46,         1;\n  5:       218790,       326503,       127249,       21190,       1720,        67,        1;\n  6:      5429970,      8323809,      3487208,      654079,      64010,      3391,       92,      1;\n  7:    156633750,    245571975,    108964365,    22373659,    2503669,    162085,     6055,    121,     1;\n  8:   5148029250,   8228462145,   3827935664,   844760636,  104745396,   7839174,   361536,  10044,   154,   1;\n  9: 189871431750, 308650887495, 149439684609, 34997119036, 4710574952, 394159794, 21192150, 732684, 15738, 191, 1;\n","recurrence_sentence":"The row polynomials satisfy (4*n-2+x)*P(n,x) = (16*n^2-4*n-6+(8*n-1)*x+x^2)*P(n-1,x) for n >= 1, with P(0,x) = 2 + x.","mathematica":"P[0, x_] := 2 + x;\nP[n_, x_] := P[n, x] = Expand[Together[((16*n^2 - 4*n - 6 + (8*n - 1)*x + x^2)*P[n - 1, x])/(4*n - 2 + x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 2 + x; for(n = 1, N, my(i = n + 1); v[i] = ((16*n^2 - 4*n - 6 + (8*n - 1)*x + x^2)*v[i - 1])/(4*n - 2 + 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-4172472abe953cea","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-0f7f0ded98411894"],"text":"A290319 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A290319 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}
