{"id":"n-1fefd5af9c8edffe","revision":1,"first_index":1,"fingerprint":"f6-d4d40bef4b30a829","name":"obreschkoff_combination of Triangle read by rows: T(n, k) = k*(n - k)*binomial(2*n+2, 2*k+1)/(4*n + 2) for 1 <= k <= n-1.","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":[["10","8"],["28","64","20"],["60","272","236","40"],["110","832","1404","656","70"],["182","2072","5764","5192","1526","112"],["280","4480","18616","27456","15496","3136","168"],["408","8736","50760","111280","102440","39744","5880","240"],["570","15744","121992","372640","508300","320960","90984","10272","330"],["770","26664","265848","1080112","2047820","1912160","881144","190608","16962","440"],["1012","42944","535788","2795584","7036232","9116352","6204184","2180288","371844","26752","572"],["1300","66352","1013012","6604312","21315112","36610112","34529992","17906512","4959812","684112","40612","728"],["1638","99008","1816100","14471600","58290050","128233280","160169240","115018400","46987850","10524800","1198340","59696","910"],["2030","143416","3112668","29772600","146435250","401662800","642022920","610086960","344934450","113919000","21063900","2013336","85358","1120"],["2480","202496","5133232","58051968","342466800","1146172800","2282580720","2773866240","2074156560","947673600","258320400","40100736","3263312","119168","1360"],["2992","279616","8187472","108081376","753390768","3022366080","7339507920","11106030240","10614744720","6411524160","2417159280","553122336","73113872","5126656","162928","1632"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["18","112","608","3072","14848","69632","319488","1441792","6422528","28311552","123731968","536870912","2315255808","9932111872","42412802048"],"nonn":true,"definitions":[{"position":0,"source":"derived:obreschkoff_combination","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["10","8"]],"recurrence":{"denominator":{"coeffs":[["0","0","0"],["2","-5","2"],["4","0","1"]]},"terms":[{"coeff":{"coeffs":[["12","-14","0","2"],["14","1","-6","7"],["4","12","1","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","8","-12","8","-4"],["0","8","-8","2","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162754,"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-01a8bab2955d58ac"]}],"region_exact":{},"incoming_edges":[{"id":"e-01a8bab2955d58ac","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-1fefd5af9c8edffe","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":"cabf9921104fbdbeb6f707c3a4f085af70330e82f614c9a77fd47081926b4827","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{}}],"premises":[{"position":0,"node_id":"n-92be85f47bd42a26","role":"sequence","name":"Triangle read by rows: T(n, k) = k*(n - k)*binomial(2*n+2, 2*k+1)/(4*n + 2) for 1 <= k <= n-1.","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-92be85f47bd42a26","role":"sequence","name":"Triangle read by rows: T(n, k) = k*(n - k)*binomial(2*n+2, 2*k+1)/(4*n + 2) for 1 <= k <= n-1.","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:   10,     8;\n   2:   28,    64,     20;\n   3:   60,   272,    236,      40;\n   4:  110,   832,   1404,     656,      70;\n   5:  182,  2072,   5764,    5192,    1526,     112;\n   6:  280,  4480,  18616,   27456,   15496,    3136,     168;\n   7:  408,  8736,  50760,  111280,  102440,   39744,    5880,     240;\n   8:  570, 15744, 121992,  372640,  508300,  320960,   90984,   10272,    330;\n   9:  770, 26664, 265848, 1080112, 2047820, 1912160,  881144,  190608,  16962,   440;\n  10: 1012, 42944, 535788, 2795584, 7036232, 9116352, 6204184, 2180288, 371844, 26752, 572;\n","recurrence_sentence":"The row polynomials satisfy (4*n^2+2*n-5*n*x+(n^2+2*n)*x^2)*P(n,x) = (4*n^2+14*n+12+(12*n^2+n-14)*x+(n^2-6*n)*x^2+(3*n^2+7*n+2)*x^3)*P(n-1,x) + ((8*n+8)*x-(8*n+12)*x^2+(2*n+8)*x^3-(2*n+4)*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 10 + 8*x.","mathematica":"P[1, x_] := 10 + 8*x;\nP[n_, x_] := P[n, x] = Expand[Together[((4*n^2 + 14*n + 12 + (12*n^2 + n - 14)*x + (n^2 - 6*n)*x^2 + (3*n^2 + 7*n + 2)*x^3)*P[n - 1, x] + ((8*n + 8)*x - (8*n + 12)*x^2 + (2*n + 8)*x^3 - (2*n + 4)*x^4)*D[P[n - 1, x], x])/(4*n^2 + 2*n - 5*n*x + (n^2 + 2*n)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 10 + 8*x; for(n = 2, N, my(i = n); v[i] = ((4*n^2 + 14*n + 12 + (12*n^2 + n - 14)*x + (n^2 - 6*n)*x^2 + (3*n^2 + 7*n + 2)*x^3)*v[i - 1] + ((8*n + 8)*x - (8*n + 12)*x^2 + (2*n + 8)*x^3 - (2*n + 4)*x^4)*deriv(v[i - 1], x))/(4*n^2 + 2*n - 5*n*x + (n^2 + 2*n)*x^2)); 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-92be85f47bd42a26","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-01a8bab2955d58ac"],"text":"A375853 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A375853 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}