{"id":"n-b41c378bbf9ce01d","revision":0,"first_index":0,"fingerprint":"f6-3bfcfaad6895e72d","name":"one_plus_derivative of theta_plus_one of Stirling2 triangle with scaled diagonals (powers of 2).","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"],["4","2"],["16","18","3"],["64","128","48","4"],["256","840","540","100","5"],["1024","5312","5280","1640","180","6"],["4096","32928","48048","22400","4060","294","7"],["16384","201728","419328","276864","73920","8736","448","8"],["65536","1226880","3564480","3218880","1191120","204624","16968","648","9"],["262144","7427072","29775360","35946880","17760960","4167072","497280","30480","900","10"],["1048576","44825088","245760768","390438400","251261120","77460768","12488784","1092960","51480","1210","11"],["4194304","270000128","2011435008","4157441024","3424227840","1351502592","283024896","33185856","2217600","82720","1584","12"],["16777216","1624197120","16363576320","43630146560","45404719360","22521827328","5964614656","900446976","80082288","4215640","127556","2028","13"],["67108864","9761964032","132541194240","452932945920","589783434240","362632478208","119148789760","22538447360","2562880320","178578400","7591584","190008","2548","14"],["268435456","58638901248","1070125068288","4663486464000","7540975303680","5687156547072","2285252825856","531154990080","75134442240","6655929280","372820448","13060320","274820","3150","15"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","6","37","244","1741","13442","111833","997416","9489145","95868158","1024429165","11537313564","136510733061","1692155605498","21919235065953"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["4","2"],["16","18","3"],["64","128","48","4"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","0","0","0","1"]]},"deriv_order":0,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["0","0","0","0","2"]]},"deriv_order":1,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["0","8","2","-2"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["0","0","4","-2"]]},"deriv_order":1,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["-8","-8"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","-4","-2"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["6","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":181341,"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-6c08c73d0ec63f1b"]},{"id":181342,"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-6c08c73d0ec63f1b"]},{"id":181343,"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-6c08c73d0ec63f1b"]}],"region_exact":{},"incoming_edges":[{"id":"e-6c08c73d0ec63f1b","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-b41c378bbf9ce01d","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":"31c193bd43c4c49d0affe6f493b3210903a6e77eed1fec7e5675034d175a2a39","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-a3fdc04d216a33db","role":"sequence","name":"theta_plus_one of Stirling2 triangle with scaled diagonals (powers of 2).","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:      4,       2;\n  2:     16,      18,        3;\n  3:     64,     128,       48,        4;\n  4:    256,     840,      540,      100,        5;\n  5:   1024,    5312,     5280,     1640,      180,       6;\n  6:   4096,   32928,    48048,    22400,     4060,     294,      7;\n  7:  16384,  201728,   419328,   276864,    73920,    8736,    448,     8;\n  8:  65536, 1226880,  3564480,  3218880,  1191120,  204624,  16968,   648,   9;\n  9: 262144, 7427072, 29775360, 35946880, 17760960, 4167072, 497280, 30480, 900, 10;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = x^4*P(n-4,x) + 2*x^4*d/dx P(n-4,x) + (8*x+2*x^2-2*x^3)*P(n-3,x) + (4*x^2-2*x^3)*d/dx P(n-3,x) - (8+8*x)*P(n-2,x) - (4*x+2*x^2)*d/dx P(n-2,x) + (6+2*x)*P(n-1,x) + 2*x*d/dx P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 4 + 2*x, P(2,x) = 16 + 18*x + 3*x^2, P(3,x) = 64 + 128*x + 48*x^2 + 4*x^3.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4 + 2*x;\nP[2, x_] := 16 + 18*x + 3*x^2;\nP[3, x_] := 64 + 128*x + 48*x^2 + 4*x^3;\nP[n_, x_] := P[n, x] = Expand[x^4*P[n - 4, x] + 2*x^4*D[P[n - 4, x], x] + (8*x + 2*x^2 - 2*x^3)*P[n - 3, x] + (4*x^2 - 2*x^3)*D[P[n - 3, x], x] - (8 + 8*x)*P[n - 2, x] - (4*x + 2*x^2)*D[P[n - 2, x], x] + (6 + 2*x)*P[n - 1, x] + 2*x*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] = 1; v[2] = 4 + 2*x; v[3] = 16 + 18*x + 3*x^2; v[4] = 64 + 128*x + 48*x^2 + 4*x^3; for(n = 4, N, my(i = n + 1); v[i] = x^4*v[i - 4] + 2*x^4*deriv(v[i - 4], x) + (8*x + 2*x^2 - 2*x^3)*v[i - 3] + (4*x^2 - 2*x^3)*deriv(v[i - 3], x) - (8 + 8*x)*v[i - 2] - (4*x + 2*x^2)*deriv(v[i - 2], x) + (6 + 2*x)*v[i - 1] + 2*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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-a7e9ebb9860a9dbe","steps":["\u03b8+1","D+1"],"edges":["e-0de2cd067f02a737","e-6c08c73d0ec63f1b"],"text":"A075497 \u2192 \u03b8+1 \u2192 D+1"},"headline":"A075497 \u2192 \u03b8+1 \u2192 D+1","chain":null}