{"id":"n-36edb24e07155eca","revision":0,"first_index":0,"fingerprint":"f6-2fbd00207b225162","name":"derivative of A triangular decomposition of the double factorial numbers A001147.","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"],["2","2"],["4","20","3"],["8","120","108","4"],["16","592","1548","464","5"],["32","2656","15504","14032","1790","6"],["64","11328","128880","256960","105600","6516","7"],["128","46976","960192","3602240","3304400","711792","22904","8"],["256","191744","6677184","42900736","73569200","36298464","4472776","78656","9"],["512","775680","44377344","458943744","1325577120","1230128928","359056320","26817216","265626","10"],["1024","3122176","286043904","4559096832","20713607040","32259165120","17943142224","3301526016","155628108","885620","11"],["2048","12531712","1805245440","42956436480","292619136000","710242251264","665196610176","237382450560","28800823320","882313000","2922788","12"],["4096","50221056","11226221568","389384376320","3839015635200","13790051463168","20084760019968","12160590879744","2921912131680","241652663600","4917022836","9565776","13"],["8192","201089024","69089955840","3429409214464","47638035289600","243796536516096","522512540884992","492971602944000","203081730507072","34052174588800","1968900012112","27049703184","31089110","14"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","1","4","27","240","2625","34020","509355","8648640","164189025","3445942500","79222218075","1979900722800","53443570205025","1549547301802500"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","0","-5"],["0","0","12"],["0","0","-4"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","2","-2"],["0","0","-4","4"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["2","-6"],["0","4"]]},"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":180791,"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-8383eb3ce6b64142"]},{"id":180792,"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-8383eb3ce6b64142"]},{"id":180793,"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-8383eb3ce6b64142"]}],"region_exact":{},"incoming_edges":[{"id":"e-8383eb3ce6b64142","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"0\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-36edb24e07155eca","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":"c40a3be6a2d7b388f6a6ef74315010c25425e5b0b150f3b0dadd2f9dbdd7a0a0","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-84fc1826f57fd756","role":"sequence","name":"A triangular decomposition of the double factorial numbers A001147.","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:   0;\n  1:   1;\n  2:   2,      2;\n  3:   4,     20,       3;\n  4:   8,    120,     108,        4;\n  5:  16,    592,    1548,      464,        5;\n  6:  32,   2656,   15504,    14032,     1790,        6;\n  7:  64,  11328,  128880,   256960,   105600,     6516,       7;\n  8: 128,  46976,  960192,  3602240,  3304400,   711792,   22904,     8;\n  9: 256, 191744, 6677184, 42900736, 73569200, 36298464, 4472776, 78656, 9;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(4*n^2-12*n+5)*x^2*P(n-2,x) - ((4*n-2)*x^2-(4*n-2)*x^3)*d/dx P(n-2,x) + (2+(4*n-6)*x)*P(n-1,x) + (2*x-2*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[-(4*n^2 - 12*n + 5)*x^2*P[n - 2, x] - ((4*n - 2)*x^2 - (4*n - 2)*x^3)*D[P[n - 2, x], x] + (2 + (4*n - 6)*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] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = -(4*n^2 - 12*n + 5)*x^2*v[i - 2] - ((4*n - 2)*x^2 - (4*n - 2)*x^3)*deriv(v[i - 2], x) + (2 + (4*n - 6)*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-54b58a0ddb0e34fe","steps":["reverse","D"],"edges":["e-5c085bd25a6bcbdf","e-8383eb3ce6b64142"],"text":"A185410 \u2192 reverse \u2192 D"},"headline":"A185410 \u2192 reverse \u2192 D","chain":null}