{"id":"n-747fc304024cb7fc","revision":0,"first_index":0,"fingerprint":"f6-354136365d8a474b","name":"derivative of basis_map_A013611 of Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)).","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":[[],["4"],["28","32"],["140","352","192"],["620","2432","2880","1024"],["2568","13728","25536","19456","5120"],["10200","69120","176448","210944","117760","24576"],["39364","323232","1051968","1729536","1510400","663552","114688"],["148756","1435328","5686464","11951104","14351360","9830400","3555328","524288"],["553220","6134176","28662720","73521152","112967680","106291200","59752448","18350080","2359296"],["2031644","25453664","137091648","415710208","780615680","939835392","724713472","344981504","92012544","10485760"],["7385424","103174080","629464704","2204675072","4900971520","7214997504","7115816960","4639424512","1913389056","450887680","46137344"],["26622960","410312448","2797252224","11118035968","28585456640","49856446464","60128051200","50105679872","28262006784","10276044800","2168455168","201326592"],["95296708","1606178496","12102140544","53834606592","157270896640","317519192064","453833375744","462357004288","333052379136","165391892480","53750005760","10267656192","872415232"],["339075340","6204205280","51202619520","252085283840","825161359360","1894524518400","3136047824896","3786714644480","3333501222912","2112293437440","936357396480","275012124672","47982837760","3758096384"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","4","60","684","6956","66408","609048","5432740","47483028","408591972","3472931516","29226323856","243935691120","2021953039876","16661184646764"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["4"]],"recurrence":{"denominator":{"coeffs":[["-44","-32","64"],["56","-16","-64"],["-12","48","0"]]},"terms":[{"coeff":{"coeffs":[["-108","-608","-1088","-512"],["316","1216","832","0"],["-96","-192","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-26","-236","-720","-832","-256"],["39","276","528","192","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":183171,"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-90d4333d9926720f"]},{"id":183172,"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-90d4333d9926720f"]},{"id":183173,"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-90d4333d9926720f"]}],"region_exact":{},"incoming_edges":[{"id":"e-90d4333d9926720f","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-747fc304024cb7fc","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":"f83d70bb1b170f92745d7699f80246166048c18341b3a02aa946af68abe9973a","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-44fd873021acce78","role":"sequence","name":"basis_map_A013611 of Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)).","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:      4;\n  2:     28,      32;\n  3:    140,     352,      192;\n  4:    620,    2432,     2880,     1024;\n  5:   2568,   13728,    25536,    19456,      5120;\n  6:  10200,   69120,   176448,   210944,    117760,     24576;\n  7:  39364,  323232,  1051968,  1729536,   1510400,    663552,   114688;\n  8: 148756, 1435328,  5686464, 11951104,  14351360,   9830400,  3555328,   524288;\n  9: 553220, 6134176, 28662720, 73521152, 112967680, 106291200, 59752448, 18350080, 2359296;\n","recurrence_sentence":"The row polynomials satisfy -(12*n^2-56*n+44-(48*n^2-16*n-32)*x+(64*n-64)*x^2)*P(n,x) = -(96*n^2-316*n+108+(192*n^2-1216*n+608)*x-(832*n-1088)*x^2+512*x^3)*P(n-1,x) + (39*n-26+(276*n-236)*x+(528*n-720)*x^2+(192*n-832)*x^3-256*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 4;\nP[n_, x_] := P[n, x] = Expand[Together[(-(96*n^2 - 316*n + 108 + (192*n^2 - 1216*n + 608)*x - (832*n - 1088)*x^2 + 512*x^3)*P[n - 1, x] + (39*n - 26 + (276*n - 236)*x + (528*n - 720)*x^2 + (192*n - 832)*x^3 - 256*x^4)*D[P[n - 1, x], x])/(-12*n^2 + 56*n - 44 + (48*n^2 - 16*n - 32)*x - (64*n - 64)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 4; for(n = 2, N, my(i = n + 1); v[i] = (-(96*n^2 - 316*n + 108 + (192*n^2 - 1216*n + 608)*x - (832*n - 1088)*x^2 + 512*x^3)*v[i - 1] + (39*n - 26 + (276*n - 236)*x + (528*n - 720)*x^2 + (192*n - 832)*x^3 - 256*x^4)*deriv(v[i - 1], x))/(-12*n^2 + 56*n - 44 + (48*n^2 - 16*n - 32)*x - (64*n - 64)*x^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-d5c14b9fec875ecc","steps":["basis A013611","D"],"edges":["e-3c5bbf704ea0dfe8","e-90d4333d9926720f"],"text":"A121574 \u2192 basis A013611 \u2192 D"},"headline":"A121574 \u2192 basis A013611 \u2192 D","chain":null}