{"id":"n-bf6741ae8e850c3e","revision":0,"first_index":0,"fingerprint":"f6-d54e0f7c6bf98dc6","name":"derivative of basis_map_A013611 of A convolution triangle of numbers based on A001906 (even-indexed Fibonacci numbers).","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"],["32","32"],["184","384","192"],["928","2976","3072","1024"],["4364","18944","29952","20480","5120"],["19648","107712","230400","240640","122880","24576"],["85872","569088","1537920","2170880","1689600","688128","114688"],["367296","2855616","9335808","16645120","17489920","10838016","3670016","524288"],["1545492","13791744","52939008","114200576","151495680","126615552","65142784","18874368","2359296"],["6419936","64675296","285170688","722498560","1159495680","1222803456","847773696","372768768","94371840","10485760"],["26393128","296268928","1475795520","4296704000","8098979840","10330177536","9032368128","5347737600","2052587520","461373440","46137344"],["107582816","1331545440","7395624960","24335226880","52719943680","78973009920","83489193984","62309007360","32180797440","10957619200","2214592512","201326592"],["435400220","5890575360","36095976960","132487536640","324456929280","558617591808","693465120768","625553178624","407061135360","186394869760","57025757184","10468982784","872415232"],["1751436160","25713957760","172331735040","698126796800","1907545456640","3712890617856","5297253384192","5610029973504","4411053047808","2543782461440","1046025863168","290514272256","48855252992","3758096384"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","4","64","760","8000","78860","745856","6856176","61726080","546964500","4786463680","41464522984","356215470784","3038825469980","25769632352000"],"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":[["-2"],["2"]]},"terms":[{"coeff":{"coeffs":[["8","8"],["4","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["3","8","4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":183021,"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-e4bceceda689d1c6"]},{"id":183022,"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-e4bceceda689d1c6"]},{"id":183023,"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-e4bceceda689d1c6"]},{"id":183024,"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-e4bceceda689d1c6"]}],"region_exact":{},"incoming_edges":[{"id":"e-e4bceceda689d1c6","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-bf6741ae8e850c3e","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":"8cd1434f3b48a9a270e3ededb8d0159fbe7437d7940842b516b1a0f962fa9fef","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-1f449c58b1930aa5","role":"sequence","name":"basis_map_A013611 of A convolution triangle of numbers based on A001906 (even-indexed Fibonacci numbers).","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:      32,       32;\n  3:     184,      384,      192;\n  4:     928,     2976,     3072,      1024;\n  5:    4364,    18944,    29952,     20480,      5120;\n  6:   19648,   107712,   230400,    240640,    122880,     24576;\n  7:   85872,   569088,  1537920,   2170880,   1689600,    688128,   114688;\n  8:  367296,  2855616,  9335808,  16645120,  17489920,  10838016,  3670016,   524288;\n  9: 1545492, 13791744, 52939008, 114200576, 151495680, 126615552, 65142784, 18874368, 2359296;\n","recurrence_sentence":"The row polynomials satisfy (2*n-2)*P(n,x) = (4*n+8+(4*n+8)*x)*P(n-1,x) + (3+8*x+4*x^2)*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[((4*n + 8 + (4*n + 8)*x)*P[n - 1, x] + (3 + 8*x + 4*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] = 4; for(n = 2, N, my(i = n + 1); v[i] = ((4*n + 8 + (4*n + 8)*x)*v[i - 1] + (3 + 8*x + 4*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-f287bed89e07609f","steps":["basis A013611","D"],"edges":["e-362ae2b737f320ab","e-e4bceceda689d1c6"],"text":"A125662 \u2192 basis A013611 \u2192 D"},"headline":"A125662 \u2192 basis A013611 \u2192 D","chain":null}