{"id":"n-ac6633a57506e279","revision":1,"first_index":1,"fingerprint":"f6-f4e4377f233f97bf","name":"obreschkoff_combination of Triangle read by rows, T(n, k) = Sum_{j=0..n} C(n-j, n-k)*E2(n, j), where E2 are the second-order Eulerian numbers A201637, for n >= 0 and 0 <= k <= n.","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","5","3"],["2","15","28","15"],["2","37","155","225","105"],["2","83","676","1960","2310","945"],["2","177","2583","13118","28980","28665","10395"],["2","367","9108","75173","274330","488565","415800","135135"],["2","749","30507","389735","2187185","6196575","9234225","6891885","2027025"],["2","1515","98764","1887446","15548940","65365300","150930780","193243050","128378250","34459425"],["2","3049","312455","8711516","101955568","607937330","2031259230","3952008060","4433779350","2653375725","654729075"],["2","6119","972940","38832651","630177548","5163348190","23886182320","66005369430","110853672930","110649213675","60235074900","13749310575"],["2","12261","2996331","168714637","3726227505","40974583650","254431887710","956105079930","2248019553780","3319258167225","2983600394775","1489508645625","316234143225"],["2","24547","9156564","719067700","21298889066","308651538195","2513950679240","12461049480880","39292037734950","80294446056825","105735253724100","86437332481500","39845502046350","7905853580625"],["2","49121","27829719","3020515306","118581564244","2232104836143","23425674437165","149734974569180","614416513480770","1665227817428175","3006648822402225","3571970059107450","2677449079305000","1146348769190625","213458046676875"],["2","98271","84258724","12548270401","646695061014","15627564244643","208349525788040","1687537443917325","8812350877600270","30746707419913485","72961521446513700","117927050035669875","127593097994612250","88300478642000625","35291730383910000","6190283353629375"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["3","10","60","524","5976","83920","1398480","26957888","589913472","14444071360","391212161280","11612030406656","374816104460544","13071033491392000","489735394015488000"],"nonn":true,"definitions":[{"position":0,"source":"derived:obreschkoff_combination","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["2","1"],["2","5","3"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","-2"],["0","1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","1","1"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0"],["0","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162792,"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-5648409316d57127"]}],"region_exact":{},"incoming_edges":[{"id":"e-5648409316d57127","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-ac6633a57506e279","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":"a44b56df56c1ca8723ffa3dab4ddadcd95a93c5d2a5fd5d0818a5a02b9162142","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{}}],"premises":[{"position":0,"node_id":"n-5b857304d1bafe7a","role":"sequence","name":"Triangle read by rows, T(n, k) = Sum_{j=0..n} C(n-j, n-k)*E2(n, j), where E2 are the second-order Eulerian numbers A201637, for n >= 0 and 0 <= k <= n.","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-5b857304d1bafe7a","role":"sequence","name":"Triangle read by rows, T(n, k) = Sum_{j=0..n} C(n-j, n-k)*E2(n, j), where E2 are the second-order Eulerian numbers A201637, for n >= 0 and 0 <= k <= n.","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,    5,      3;\n   3: 2,   15,     28,      15;\n   4: 2,   37,    155,     225,       105;\n   5: 2,   83,    676,    1960,      2310,       945;\n   6: 2,  177,   2583,   13118,     28980,     28665,      10395;\n   7: 2,  367,   9108,   75173,    274330,    488565,     415800,     135135;\n   8: 2,  749,  30507,  389735,   2187185,   6196575,    9234225,    6891885,    2027025;\n   9: 2, 1515,  98764, 1887446,  15548940,  65365300,  150930780,  193243050,  128378250,   34459425;\n  10: 2, 3049, 312455, 8711516, 101955568, 607937330, 2031259230, 3952008060, 4433779350, 2653375725, 654729075;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+(n-2)*x)*P(n-2,x) + (x+x^2)*d/dx P(n-2,x) + n*x*P(n-1,x) + (x+x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 2 + x, P(2,x) = 2 + 5*x + 3*x^2.","mathematica":"P[1, x_] := 2 + x;\nP[2, x_] := 2 + 5*x + 3*x^2;\nP[n_, x_] := P[n, x] = Expand[(1 + (n - 2)*x)*P[n - 2, x] + (x + x^2)*D[P[n - 2, x], x] + n*x*P[n - 1, x] + (x + x^2)*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; v[2] = 2 + 5*x + 3*x^2; for(n = 3, N, my(i = n); v[i] = (1 + (n - 2)*x)*v[i - 2] + (x + x^2)*deriv(v[i - 2], x) + n*x*v[i - 1] + (x + 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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-5b857304d1bafe7a","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-5648409316d57127"],"text":"A112493 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A112493 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}