{"id":"n-43121871192f6f93","revision":1,"first_index":0,"fingerprint":"f6-85cf5e3438332e7d","name":"obreschkoff_combination of Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 2*x + 5.","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":[["6","5"],["30","75","25"],["150","785","740","125"],["750","7095","13585","6505","625"],["3750","59165","198880","192390","53410","3125"],["18750","469155","2546305","4278230","2413080","419495","15625"],["93750","3596585","29848500","79790475","78058550","27904575","3195840","78125"],["468750","26926095","328945705","1320702135","2043852885","1272492085","304637715","23824005","390625"],["2343750","198170165","3465746720","20064824060","46229223020","45506072270","19121277680","3188035100","174814910","1953125"],["11718750","1440316155","35305802185","285959337340","939591900520","1372093120090","914150652310","270303806380","32315937250","1267844995","9765625"],["58593750","10371275585","350530642660","3880645978185","17610282475860","36465936973410","36033435753840","16982091160890","3645647973990","319655021565","9115149340","48828125"],["292968750","74161429095","3411525889825","50685027515285","309892958693745","879793759978230","1229833672259850","861370606700250","296791513511040","47389910183955","3102242143845","65104873505","244140625"],["1464843750","527528441165","32687880826560","642186671654370","5187137082948190","19671228628954635","37552977658160400","37202937745878780","19119750980948130","4941030596952435","598208287360240","29659800111810","462716536410","1220703125"],["7324218750","3737620963155","309359140142065","7940439896994930","83393305827577200","413786226328027865","1049807110086661095","1417973001542068140","1032275400969736950","399853612430027925","79094461902279135","7374919490461250","280226868471420","3276369270495","6103515625"],["36621093750","26402604554585","2899173566344820","96257064442931255","1297314622676166490","8281838122458051625","27326150931070083280","48914719776937533135","48546553885482126090","26707661756428374195","7965377815855600900","1226160600670882285","89191504092612070","2622684318836555","23133559502840","30517578125"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["11","130","1800","28560","510720","10160640","222566400","5322240000","137952460800","3852450201600","115307819827200","3682411020288000","124978798264320000","4492095091900416000","170456797541302272000"],"nonn":true,"definitions":[{"position":0,"source":"derived:obreschkoff_combination","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["6","5"],["30","75","25"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["5","1"],["0","2"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","2","-2"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["4","5"],["0","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162850,"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-682ccea737887217"]}],"region_exact":{},"incoming_edges":[{"id":"e-682ccea737887217","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-43121871192f6f93","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":"55c76e58b4ab379d9485873407cfbd2e038d8a7e314e694e242d2a2b83e37c71","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-3f0a1785078e99be","role":"sequence","name":"Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 2*x + 5.","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-3f0a1785078e99be","role":"sequence","name":"Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 2*x + 5.","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:        6,          5;\n  1:       30,         75,          25;\n  2:      150,        785,         740,          125;\n  3:      750,       7095,       13585,         6505,          625;\n  4:     3750,      59165,      198880,       192390,        53410,          3125;\n  5:    18750,     469155,     2546305,      4278230,      2413080,        419495,        15625;\n  6:    93750,    3596585,    29848500,     79790475,     78058550,      27904575,      3195840,        78125;\n  7:   468750,   26926095,   328945705,   1320702135,   2043852885,    1272492085,    304637715,     23824005,      390625;\n  8:  2343750,  198170165,  3465746720,  20064824060,  46229223020,   45506072270,  19121277680,   3188035100,   174814910,    1953125;\n  9: 11718750, 1440316155, 35305802185, 285959337340, 939591900520, 1372093120090, 914150652310, 270303806380, 32315937250, 1267844995, 9765625;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (5+(2*n+1)*x)*P(n-2,x) + (2*x-2*x^2)*d/dx P(n-2,x) + (4+(2*n+5)*x)*P(n-1,x) + (2*x-2*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 6 + 5*x, P(1,x) = 30 + 75*x + 25*x^2.","mathematica":"P[0, x_] := 6 + 5*x;\nP[1, x_] := 30 + 75*x + 25*x^2;\nP[n_, x_] := P[n, x] = Expand[(5 + (2*n + 1)*x)*P[n - 2, x] + (2*x - 2*x^2)*D[P[n - 2, x], x] + (4 + (2*n + 5)*x)*P[n - 1, x] + (2*x - 2*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] = 6 + 5*x; v[2] = 30 + 75*x + 25*x^2; for(n = 2, N, my(i = n + 1); v[i] = (5 + (2*n + 1)*x)*v[i - 2] + (2*x - 2*x^2)*deriv(v[i - 2], x) + (4 + (2*n + 5)*x)*v[i - 1] + (2*x - 2*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-3f0a1785078e99be","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-682ccea737887217"],"text":"A257615 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A257615 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}