{"id":"n-f4f552e83b5e543f","revision":0,"first_index":0,"fingerprint":"f6-b523dc1f3adb9012","name":"derivative of basis_map_A007318 of Riordan array (1/((1-x)(1-3x)),x/((1-x)(1-3x))).","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"],["10","2"],["69","30","3"],["410","282","60","4"],["2252","2140","714","100","5"],["11790","14358","6600","1440","150","6"],["59767","88950","52395","15700","2535","210","7"],["296050","521210","375600","144040","31850","4074","280","8"],["1441155","2931300","2502630","1173900","331695","57960","6132","360","9"],["6920900","15975180","15783300","8774220","3030300","675612","97440","8784","450","10"],["32876792","84923800","95371380","61393800","25124190","6840540","1257018","154200","12105","550","11"],["154790220","442447044","556892700","407808180","193272300","62324892","13973400","2181240","232650","16170","660","12"],["723378253","2267044260","3162019926","2597383100","1400749155","523029600","138808320","26413200","3580830","337700","21054","780","13"],["3359301370","11454636674","17539987920","15979517528","9670657750","4109072310","1262322600","284160360","46921050","5618690","474760","26832","910","14"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","1","12","102","756","5211","34344","219564","1373112","8445141","51266196","307954386","1833939468","10842766191","63712698768"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["1"]],"recurrence":{"denominator":{"coeffs":[["-2"],["2"]]},"terms":[{"coeff":{"coeffs":[["10","2"],["5","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["13","10","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":179867,"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-ad207e236e5af4a7"]},{"id":179868,"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-ad207e236e5af4a7"]},{"id":179869,"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-ad207e236e5af4a7"]},{"id":179870,"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-ad207e236e5af4a7"]}],"region_exact":{},"incoming_edges":[{"id":"e-ad207e236e5af4a7","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-f4f552e83b5e543f","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":"931f4ef54249126851e35058c688ea47e142d291e4f4f82b82fe00ff80c6eb25","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-317113238d23620b","role":"sequence","name":"basis_map_A007318 of Riordan array (1/((1-x)(1-3x)),x/((1-x)(1-3x))).","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:      10,       2;\n  3:      69,      30,       3;\n  4:     410,     282,      60,       4;\n  5:    2252,    2140,     714,     100,      5;\n  6:   11790,   14358,    6600,    1440,    150,     6;\n  7:   59767,   88950,   52395,   15700,   2535,   210,    7;\n  8:  296050,  521210,  375600,  144040,  31850,  4074,  280,   8;\n  9: 1441155, 2931300, 2502630, 1173900, 331695, 57960, 6132, 360, 9;\n","recurrence_sentence":"The row polynomials satisfy (2*n-2)*P(n,x) = (5*n+10+(n+2)*x)*P(n-1,x) + (13+10*x+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[Together[((5*n + 10 + (n + 2)*x)*P[n - 1, x] + (13 + 10*x + x^2)*D[P[n - 1, x], x])/(2*n - 2)]];\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] = ((5*n + 10 + (n + 2)*x)*v[i - 1] + (13 + 10*x + x^2)*deriv(v[i - 1], x))/(2*n - 2)); 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-6466a8550c08a52f","steps":["basis A007318","D"],"edges":["e-235ea1c8ecbb7de6","e-ad207e236e5af4a7"],"text":"A116414 \u2192 basis A007318 \u2192 D"},"headline":"A116414 \u2192 basis A007318 \u2192 D","chain":null}