{"id":"n-cf8b34d3478a2380","revision":0,"first_index":0,"fingerprint":"f6-5007d7917df89719","name":"derivative of basis_map_A013610 of Triangle read by rows, s_4(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 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":[[],["3"],["36","18"],["528","432","81"],["9600","10080","3240","324"],["210432","259200","110160","19440","1215"],["5419008","7483392","3810240","907200","102060","4374"],["160579584","242049024","140361984","40642560","6259680","489888","15309"],["5386272768","8709046272","5580804096","1862009856","352719360","38211264","2204496","52488"],["201833054208","345843302400","240099655680","89355571200","19680433920","2645395200","213101280","9447840","177147"],["8358702612480","15044730421248","11160281088000","4534621470720","1122300748800","176665370112","17782934400","1108546560","38972340","590490"],["379198132715520","712120354209792","558753654767616","244164469063680","66386063462400","11813558980608","1400777908992","109746109440","5456127600","155889360","1948617"],["18703779952066560","36456965798363136","30024717022789632","13954909134127104","4102147805184000","806040659533824","108584796063744","10070480369664","632287244160","25669781280","607968504","6377292"],["996705497430097920","2007984901227282432","1725341631277694976","845754143065767936","265642595122544640","56681846394421248","8467551703683072","895955708583936","66866807943936","3442452773760","116324640432","2321334288","20726199"],["57069172732062597120","118427387453308403712","105655063407073689600","54263598536793784320","18047563363359129600","4130496740528750592","672569357786726400","79205125521899520","6768391761331200","415666715143680","17874274017600","510693543360","8705003580","66961566"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","3","54","1041","23244","600447","17726274","590398029","21931320600","899680138875","40416232755150","1973952371183625","104167874219868900","5907544549857494775","358352259168650981850"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["3"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-16","24","-9"],["32","-24","0"],["-16","0","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-4","6"],["8","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":190601,"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-0590da0e1b621364"]},{"id":190602,"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-0590da0e1b621364"]},{"id":190603,"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-0590da0e1b621364"]},{"id":190604,"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-0590da0e1b621364"]}],"region_exact":{},"incoming_edges":[{"id":"e-0590da0e1b621364","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-cf8b34d3478a2380","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":"a5fcf3b0ecda97747359c926aa686cebedfa19deaef79cf1998e41607b752422","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-a06f8ee83f308dcc","role":"sequence","name":"basis_map_A013610 of Triangle read by rows, s_4(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.","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:            3;\n  2:           36,           18;\n  3:          528,          432,           81;\n  4:         9600,        10080,         3240,         324;\n  5:       210432,       259200,       110160,       19440,        1215;\n  6:      5419008,      7483392,      3810240,      907200,      102060,       4374;\n  7:    160579584,    242049024,    140361984,    40642560,     6259680,     489888,     15309;\n  8:   5386272768,   8709046272,   5580804096,  1862009856,   352719360,   38211264,   2204496,   52488;\n  9: 201833054208, 345843302400, 240099655680, 89355571200, 19680433920, 2645395200, 213101280, 9447840, 177147;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(16*n^2-32*n+16+(24*n-24)*x+9*x^2)*P(n-2,x) + (8*n-4+6*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 3;\nP[n_, x_] := P[n, x] = Expand[-(16*n^2 - 32*n + 16 + (24*n - 24)*x + 9*x^2)*P[n - 2, x] + (8*n - 4 + 6*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 3; for(n = 2, N, my(i = n + 1); v[i] = -(16*n^2 - 32*n + 16 + (24*n - 24)*x + 9*x^2)*v[i - 2] + (8*n - 4 + 6*x)*v[i - 1]); 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-9c5c208df8605708","steps":["basis A013610","D"],"edges":["e-bf3b7a758db4c427","e-0590da0e1b621364"],"text":"A225471 \u2192 basis A013610 \u2192 D"},"headline":"A225471 \u2192 basis A013610 \u2192 D","chain":null}