{"id":"n-7423cb1a102aa138","revision":0,"first_index":0,"fingerprint":"f6-4979d70c29ca7b2b","name":"derivative of basis_map_A013609 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":[[],["2"],["24","8"],["352","192","24"],["6400","4480","960","64"],["140288","115200","32640","3840","160"],["3612672","3325952","1128960","179200","13440","384"],["107053056","107577344","41588736","8028160","824320","43008","896"],["3590848512","3870687232","1653571584","367804416","46448640","3354624","129024","2048"],["134555369472","153708134400","71140638720","17650483200","2591662080","232243200","12472320","368640","4608"],["5572468408320","6686546853888","3306749952000","895727697920","147792691200","15509716992","1040793600","43253760","1013760","10240"],["252798755143680","316497935204352","165556638449664","48230018580480","8742197657600","1037130006528","81984258048","4282122240","141926400","2703360","22528"],["12469186634711040","16203095910383616","8896212451196928","2756525261062144","540200534016000","70763514691584","6355214401536","392934457344","16447242240","445153280","7028736","49152"],["664470331620065280","892437733878792192","511212335193391104","167062546778423296","34981740921487360","4976184045600768","495586016493568","34958796128256","1739359125504","59697397760","1344831488","17891328","106496"],["38046115154708398080","52634394423692623872","31305203972466278400","10718735513440747520","2376633858549350400","362622484765212672","39363913030041600","3090460620881920","176061402316800","7208296775680","206644838400","3936092160","44728320","229376"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","2","32","568","11904","292128","8260608","265115520","9532846080","379891376640","16625880391680","792949086074880","40942749354393600","2275673217669734400","135486343261598515200"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["2"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-16","16","-4"],["32","-16","0"],["-16","0","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-4","4"],["8","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":189842,"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-030a09165e82c067"]},{"id":189843,"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-030a09165e82c067"]},{"id":189844,"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-030a09165e82c067"]},{"id":189845,"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-030a09165e82c067"]}],"region_exact":{},"incoming_edges":[{"id":"e-030a09165e82c067","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-7423cb1a102aa138","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":"5266c546b1e7be1e9dc565c379ae25e7838797a6939a577db30bd403d15238ee","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-bf0f86dc9ba61887","role":"sequence","name":"basis_map_A013609 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":[{"id":"l-a772ce0092abd174","kind":"fingerprint_related","node_a_id":"n-8403a18b81fa8c5b","node_b_id":"n-7423cb1a102aa138","fingerprint":"f6-4979d70c29ca7b2b","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4979d70c29ca7b2b"},"other_id":"n-8403a18b81fa8c5b","other":{"id":"n-8403a18b81fa8c5b","name":"derivative of basis_map_A007318 of Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.","first_index":0,"fingerprint":"f6-4979d70c29ca7b2b","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["1"],["6","2"]],"card_rows":[["1"],["6","2"],["44","24","3"],["400","280","60","4"]],"row_sums_preview":["0","1","8","71","744","9129","129072","2071215"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4f6e747b794bcc2e","steps":["x \u21a6 x+1","D"],"edges":["e-29cb0c45f74a7f15","e-13269c16224e26df"],"text":"A028338 \u2192 x \u21a6 x+1 \u2192 D"},"headline":"A028338 \u2192 x \u21a6 x+1 \u2192 D","chain":null}}],"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:            2;\n  2:           24,            8;\n  3:          352,          192,          24;\n  4:         6400,         4480,         960,          64;\n  5:       140288,       115200,       32640,        3840,        160;\n  6:      3612672,      3325952,     1128960,      179200,      13440,       384;\n  7:    107053056,    107577344,    41588736,     8028160,     824320,     43008,      896;\n  8:   3590848512,   3870687232,  1653571584,   367804416,   46448640,   3354624,   129024,   2048;\n  9: 134555369472, 153708134400, 71140638720, 17650483200, 2591662080, 232243200, 12472320, 368640, 4608;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(16*n^2-32*n+16+(16*n-16)*x+4*x^2)*P(n-2,x) + (8*n-4+4*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[-(16*n^2 - 32*n + 16 + (16*n - 16)*x + 4*x^2)*P[n - 2, x] + (8*n - 4 + 4*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] = 2; for(n = 2, N, my(i = n + 1); v[i] = -(16*n^2 - 32*n + 16 + (16*n - 16)*x + 4*x^2)*v[i - 2] + (8*n - 4 + 4*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 A013609","D"],"edges":["e-8d6d917622b49386","e-030a09165e82c067"],"text":"A225471 \u2192 basis A013609 \u2192 D"},"headline":"A225471 \u2192 basis A013609 \u2192 D","chain":null}