{"id":"n-776206235fcb89e5","revision":0,"first_index":0,"fingerprint":"f6-0e58c6c76130f194","name":"derivative of basis_map_A007318 of Triangle read by rows, s_3(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":[[],["1"],["9","2"],["99","36","3"],["1350","630","90","4"],["22194","12150","2295","180","5"],["428652","263088","59535","6300","315","6"],["9526572","6382152","1644867","211680","14490","504","7"],["239660208","172224792","49050036","7273476","612360","29484","756","8"],["6735365136","5129389800","1582688160","261783900","25625565","1530900","54810","1080","9"],["209203521120","167352424272","55174729500","9963767880","1095996825","76677678","3430350","95040","1485","10"],["7117993208160","5941037043216","2071794709908","402370646040","48622605075","3845559564","202658841","7056720","155925","1980","11"],["263318896863360","228113319972096","83496165121512","17247736203984","2253377090250","196787270394","11782204434","485652024","13552110","244530","2574","12"],["10523998066976640","9423057272638464","3598520419697112","783989932117392","109441376774730","10378756053666","689091121719","32405805432","1074890817","24594570","369369","3276","13"],["451935801295937280","416816401584768768","165272443536236400","37725604375193280","5576525239685400","567237406383648","41050375841475","2148576538680","81601945425","2227295070","42567525","540540","4095","14"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","1","11","138","2074","36824","757896","17780272","468851120","13736439360","442870644160","15585873645440","594638564177280","24450108421043200","1077937296262937600"],"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":[["-9","6","-1"],["18","-6","0"],["-9","0","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-3","2"],["6","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":187384,"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-2cf677ac5d418c9c"]},{"id":187385,"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-2cf677ac5d418c9c"]},{"id":187386,"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-2cf677ac5d418c9c"]},{"id":187387,"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-2cf677ac5d418c9c"]}],"region_exact":{},"incoming_edges":[{"id":"e-2cf677ac5d418c9c","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-776206235fcb89e5","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":"769c626c3eb0e1f97aecd1de6ca33432a6065f14ccf6ce17997e43dde60ed601","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-b6cbf920aaa38713","role":"sequence","name":"basis_map_A007318 of Triangle read by rows, s_3(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:          1;\n  2:          9,          2;\n  3:         99,         36,          3;\n  4:       1350,        630,         90,         4;\n  5:      22194,      12150,       2295,       180,        5;\n  6:     428652,     263088,      59535,      6300,      315,       6;\n  7:    9526572,    6382152,    1644867,    211680,    14490,     504,     7;\n  8:  239660208,  172224792,   49050036,   7273476,   612360,   29484,   756,    8;\n  9: 6735365136, 5129389800, 1582688160, 261783900, 25625565, 1530900, 54810, 1080, 9;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(9*n^2-18*n+9+(6*n-6)*x+x^2)*P(n-2,x) + (6*n-3+2*x)*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[-(9*n^2 - 18*n + 9 + (6*n - 6)*x + x^2)*P[n - 2, x] + (6*n - 3 + 2*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] = 1; for(n = 2, N, my(i = n + 1); v[i] = -(9*n^2 - 18*n + 9 + (6*n - 6)*x + x^2)*v[i - 2] + (6*n - 3 + 2*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-8229dbdd6a4b5cd9","steps":["x \u21a6 x+1","D"],"edges":["e-64ce901fd869872f","e-2cf677ac5d418c9c"],"text":"A225470 \u2192 x \u21a6 x+1 \u2192 D"},"headline":"A225470 \u2192 x \u21a6 x+1 \u2192 D","chain":null}