{"id":"n-778c27541fff7d28","revision":0,"first_index":0,"fingerprint":"f6-c0dc793f2abea0c9","name":"one_plus_derivative of Triangle of 5-Eulerian numbers.","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":[["1"],["6","5"],["17","66","25"],["40","381","546","125"],["87","1658","5364","4026","625"],["182","6273","37864","60730","27906","3125"],["373","21906","223143","641380","604827","186186","15625"],["756","72797","1176234","5498355","9045296","5537091","1210866","78125"],["1523","234258","5757850","41154470","107595684","112675150","47782574","7733946","390625"],["3058","737793","26773440","280391664","1096392780","1796198082","1283661792","394865640","48747426","1953125"],["6129","2289818","119991381","1784853768","10002152034","24154411596","26700160242","13677573864","3158025837","304203306","9765625"],["12272","7034925","523441426","10801088889","84049013304","286757227722","462559145580","363399606954","138402644616","24626229001","1883813586","48828125"],["24559","21460650","2237914176","62893414706","663306848379","3097280822292","6990042076584","7955827824564","4616652946149","1344632115266","188255844984","11595850266","244140625"],["49134","65143265","9423643656","355459773270","4984650928150","31081277698227","95066191149120","150348747809028","125724807707130","55510126643895","12644414843704","1416539991750","71039945346","1220703125"],["98285","197051202","39225121987","1962511185676","36032543335365","294209252888390","1189671109255731","2532525537135816","2928269875069023","1855799555086510","638289054446625","115797886966892","10524143544047","433563890826","6103515625"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","11","108","1092","11760","136080","1693440","22619520","323326080","4929724800","79913433600","1373058086400","24932991283200","477213966028800","9603560558592000"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["6","5"]],"recurrence":{"denominator":{"coeffs":[["2","2"],["1","0"]]},"terms":[{"coeff":{"coeffs":[["-6","-13","-18","-17","-6"],["-2","-9","-20","-14","-2"],["0","-2","-7","-3","0"],["0","0","-1","0","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","-3","-5","1","5","2"],["0","-1","-4","2","3","0"],["0","0","-1","1","0","0"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["7","16","11","2"],["3","9","5","0"],["0","2","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2","0","-2"],["0","1","-1","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":185068,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-4704c1422914f748"]},{"id":185069,"position":1,"claim":"one_signed_coefficients","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":0},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 0: the coefficients of P_n have one sign","region_exact":null,"edges":["e-4704c1422914f748"]},{"id":185070,"position":2,"claim":"hadamard_kernel","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":0},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-4704c1422914f748"]},{"id":185071,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-4704c1422914f748"]}],"region_exact":{},"incoming_edges":[{"id":"e-4704c1422914f748","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-778c27541fff7d28","premise_count":1,"theorem":"Laguerre's theorem on differential operators: if phi is a real polynomial with only real roots, then phi(D) maps every real-rooted polynomial to a real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"8e393450c4094c4234e9b34ae1f5e739e95ea73001b62b728770eeeb9deb6182","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-35f03041aed96954","role":"sequence","name":"Triangle of 5-Eulerian numbers.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"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:    1;\n  1:    6,      5;\n  2:   17,     66,       25;\n  3:   40,    381,      546,       125;\n  4:   87,   1658,     5364,      4026,        625;\n  5:  182,   6273,    37864,     60730,      27906,       3125;\n  6:  373,  21906,   223143,    641380,     604827,     186186,      15625;\n  7:  756,  72797,  1176234,   5498355,    9045296,    5537091,    1210866,     78125;\n  8: 1523, 234258,  5757850,  41154470,  107595684,  112675150,   47782574,   7733946,   390625;\n  9: 3058, 737793, 26773440, 280391664, 1096392780, 1796198082, 1283661792, 394865640, 48747426, 1953125;\n","recurrence_sentence":"The row polynomials satisfy (n+2+2*x)*P(n,x) = -(2*n+6+(2*n^2+9*n+13)*x+(n^3+7*n^2+20*n+18)*x^2+(3*n^2+14*n+17)*x^3+(2*n+6)*x^4)*P(n-2,x) - ((n+3)*x+(n^2+4*n+5)*x^2-(n^2+2*n+1)*x^3-(3*n+5)*x^4-2*x^5)*d/dx P(n-2,x) + (3*n+7+(2*n^2+9*n+16)*x+(5*n+11)*x^2+2*x^3)*P(n-1,x) + ((n+2)*x-n*x^2-2*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 6 + 5*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 6 + 5*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(2*n + 6 + (2*n^2 + 9*n + 13)*x + (n^3 + 7*n^2 + 20*n + 18)*x^2 + (3*n^2 + 14*n + 17)*x^3 + (2*n + 6)*x^4)*P[n - 2, x] - ((n + 3)*x + (n^2 + 4*n + 5)*x^2 - (n^2 + 2*n + 1)*x^3 - (3*n + 5)*x^4 - 2*x^5)*D[P[n - 2, x], x] + (3*n + 7 + (2*n^2 + 9*n + 16)*x + (5*n + 11)*x^2 + 2*x^3)*P[n - 1, x] + ((n + 2)*x - n*x^2 - 2*x^3)*D[P[n - 1, x], x])/(n + 2 + 2*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 6 + 5*x; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n + 6 + (2*n^2 + 9*n + 13)*x + (n^3 + 7*n^2 + 20*n + 18)*x^2 + (3*n^2 + 14*n + 17)*x^3 + (2*n + 6)*x^4)*v[i - 2] - ((n + 3)*x + (n^2 + 4*n + 5)*x^2 - (n^2 + 2*n + 1)*x^3 - (3*n + 5)*x^4 - 2*x^5)*deriv(v[i - 2], x) + (3*n + 7 + (2*n^2 + 9*n + 16)*x + (5*n + 11)*x^2 + 2*x^3)*v[i - 1] + ((n + 2)*x - n*x^2 - 2*x^3)*deriv(v[i - 1], x))/(n + 2 + 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-35f03041aed96954","steps":["D+1"],"edges":["e-4704c1422914f748"],"text":"A144699 \u2192 D+1"},"headline":"A144699 \u2192 D+1","chain":null}