{"id":"n-5254037deeca106e","revision":1,"first_index":0,"fingerprint":"f6-a71ac548f77bbfdd","name":"obreschkoff_combination of Triangle read by rows, coefficients of the polynomials P(k, x) = (1/2) Sum_{p=0..k-1} Stirling2(k, p+1)*x^p*(1-4*x)^(k-1-p)*(2*p+2)!/(p+1)!.","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":[["2","2"],["2","12","4"],["2","40","76","8"],["2","104","588","472","16"],["2","240","3100","7480","2880","32"],["2","520","13324","71256","87344","17408","64"],["2","1088","50748","514520","1406240","966432","104768","128"],["2","2232","179116","3131576","16035600","25148032","10328192","629376","256"],["2","4528","601372","17023288","147271552","434241952","420573568","107898112","3778048","512"],["2","9128","1952076","85577880","1168238064","5786570496","10663574208","6710320896","1110466560","22672384","1024"],["2","18336","6190972","406833048","8351158752","64563381792","200309398464","243985502592","103509616128","11316278272","136043520","2048"],["2","36760","19318636","1856629880","55303173840","633772331904","3059307605760","6317677890432","5296913383680","1557870653440","114568573952","816281600","4096"],["2","73616","59597916","8219097592","345725416000","5653679988384","40191575780352","129323196175872","185711604240384","110513535731200","23025269036032","1154936948736","4897734656","8192"],["2","147336","182376460","35559424600","2067745047984","46867694956544","471267808156992","2228855837091840","5009486356976640","5170012608775168","2236335294369792","335755279763456","11609774919680","29386506240","16384"],["2","294784","554889148","151176599640","11947666512032","366860510872352","5059530687404608","33704662876676736","111458080722600960","181224702994008064","137904097796384768","44191455309799424","4846884507967488","116489569361920","176319250432","32768"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["4","18","126","1182","13734","189918","3043926","55454382","1131392934","25549382718","632584423926","17038105883982","495932699828934","15512323528529118","518885040692655126"],"nonn":true,"definitions":[{"position":0,"source":"derived:obreschkoff_combination","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["2","2"],["2","12","4"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","-6"],["0","4"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","1","-4"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","2"],["0","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162803,"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-08a4b5e6d1540539"]}],"region_exact":{},"incoming_edges":[{"id":"e-08a4b5e6d1540539","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-5254037deeca106e","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":"dd733c8b717c370e39f331c60c311cf1e335c9ab81c227eaf9efa05417ec9180","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-633aaaf6fe98dc51","role":"sequence","name":"Triangle read by rows, coefficients of the polynomials P(k, x) = (1/2) Sum_{p=0..k-1} Stirling2(k, p+1)*x^p*(1-4*x)^(k-1-p)*(2*p+2)!/(p+1)!.","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-633aaaf6fe98dc51","role":"sequence","name":"Triangle read by rows, coefficients of the polynomials P(k, x) = (1/2) Sum_{p=0..k-1} Stirling2(k, p+1)*x^p*(1-4*x)^(k-1-p)*(2*p+2)!/(p+1)!.","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,    2;\n  1: 2,   12,       4;\n  2: 2,   40,      76,        8;\n  3: 2,  104,     588,      472,         16;\n  4: 2,  240,    3100,     7480,       2880,         32;\n  5: 2,  520,   13324,    71256,      87344,      17408,          64;\n  6: 2, 1088,   50748,   514520,    1406240,     966432,      104768,        128;\n  7: 2, 2232,  179116,  3131576,   16035600,   25148032,    10328192,     629376,        256;\n  8: 2, 4528,  601372, 17023288,  147271552,  434241952,   420573568,  107898112,    3778048,      512;\n  9: 2, 9128, 1952076, 85577880, 1168238064, 5786570496, 10663574208, 6710320896, 1110466560, 22672384, 1024;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+(4*n-6)*x)*P(n-2,x) + (x-4*x^2)*d/dx P(n-2,x) + (4*n+2)*x*P(n-1,x) + (x-4*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 2 + 2*x, P(1,x) = 2 + 12*x + 4*x^2.","mathematica":"P[0, x_] := 2 + 2*x;\nP[1, x_] := 2 + 12*x + 4*x^2;\nP[n_, x_] := P[n, x] = Expand[(1 + (4*n - 6)*x)*P[n - 2, x] + (x - 4*x^2)*D[P[n - 2, x], x] + (4*n + 2)*x*P[n - 1, x] + (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] = 2 + 2*x; v[2] = 2 + 12*x + 4*x^2; for(n = 2, N, my(i = n + 1); v[i] = (1 + (4*n - 6)*x)*v[i - 2] + (x - 4*x^2)*deriv(v[i - 2], x) + (4*n + 2)*x*v[i - 1] + (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-633aaaf6fe98dc51","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-08a4b5e6d1540539"],"text":"A142963 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A142963 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}