{"id":"n-9ae14afeef916271","revision":0,"first_index":1,"fingerprint":"f6-bc765d2be295a474","name":"one_plus_derivative of Triangle read by rows, the Bell transform of the triple factorial numbers A007559(n+1) without column 0.","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":[["1"],["5","1"],["40","14","1"],["440","208","27","1"],["6160","3560","640","44","1"],["104720","70240","15720","1520","65","1"],["2094400","1579760","416080","50680","3080","90","1"],["48171200","40008640","11981760","1733760","133840","5600","119","1"],["1252451200","1128404480","375497920","62260800","5748960","307440","9408","152","1"],["36321084800","35104585600","12768840000","2367843520","251976480","16201920","636720","14880","189","1"],["1162274713600","1194772409600","469245145600","95643363200","11438996800","851447520","40397280","1217040","22440","230","1"],["40679614976000","44174081459200","18556842796800","4103692947200","541764238400","45496897600","2507736000","91512960","2181960","32560","275","1"],["1545825369088000","1763468791884800","786447793331200","186808253632000","26857790960000","2498130835200","155837281600","6625819200","191883120","3712280","45760","324","1"],["63378840132608000","75611514628249600","35580723905126400","9006010953139200","1395408846832000","141771809379200","9822483070400","473876603200","16028092080","377497120","6046040","62608","377","1"],["2788668965834752000","3465896998403840000","1712324814264064000","458863238702668800","75985635601075200","8342227506412800","632895962899200","33990189833600","1307337391360","36035439440","703983280","9489480","83720","434","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","6","55","676","10405","192266","4144091","102034920","2824680361","86831184110","2934267713311","108104094778956","4312068788473485","185124583046706226","8510750110551933315"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-1","1"],["3","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":188734,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-5d9d30a1dbf4de00"]},{"id":188735,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-5d9d30a1dbf4de00"]},{"id":188736,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-5d9d30a1dbf4de00"]},{"id":188737,"position":3,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2","region_exact":null,"edges":["e-5d9d30a1dbf4de00"]}],"region_exact":{},"incoming_edges":[{"id":"e-5d9d30a1dbf4de00","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-9ae14afeef916271","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":"2f2188121f4aaa1b162461b53dd2b34d78f030e8a442514a2bc7531cb7f7b910","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-df97cd413ec843e8","role":"sequence","name":"Triangle read by rows, the Bell transform of the triple factorial numbers A007559(n+1) without column 0.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:           1;\n   2:           5,           1;\n   3:          40,          14,           1;\n   4:         440,         208,          27,          1;\n   5:        6160,        3560,         640,         44,         1;\n   6:      104720,       70240,       15720,       1520,        65,        1;\n   7:     2094400,     1579760,      416080,      50680,      3080,       90,      1;\n   8:    48171200,    40008640,    11981760,    1733760,    133840,     5600,    119,     1;\n   9:  1252451200,  1128404480,   375497920,   62260800,   5748960,   307440,   9408,   152,   1;\n  10: 36321084800, 35104585600, 12768840000, 2367843520, 251976480, 16201920, 636720, 14880, 189, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3*n-1+x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(3*n - 1 + x)*P[n - 1, x] + x*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] = 1; for(n = 2, N, my(i = n); v[i] = (3*n - 1 + x)*v[i - 1] + x*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-df97cd413ec843e8","steps":["D+1"],"edges":["e-5d9d30a1dbf4de00"],"text":"A035469 \u2192 D+1"},"headline":"A035469 \u2192 D+1","chain":null}