{"id":"n-caba3b00effe5aa9","revision":0,"first_index":0,"fingerprint":"f6-1e104af877c3f00c","name":"derivative of reversal of Bisection of Fibonacci triangle A037027: even-indexed members of column sequences of A037027 (not counting leading zeros).","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"],["5","10"],["9","40","39"],["14","102","213","136"],["20","210","699","940","445"],["27","380","1782","3768","3720","1398"],["35","630","3885","11440","17610","13710","4270"],["44","980","7602","29140","62010","74838","48055","12776"],["54","1452","13734","65628","180610","299778","297129","162272","37629"],["65","2070","23328","134820","459795","982380","1333990","1120872","532395","109460"],["77","2860","37719","257796","1057595","2788338","4856845","5574296","4062969","1707110","315227"],["90","3850","58575","465300","2246200","7095900","15202950","22320800","22169700","14263800","5372400","900300"],["104","5070","87945","800800","4472325","16564860","42397950","76372560","96822450","84722800","48778620","16646700","2553434"],["119","6552","128310","1324180","8440575","36045360","107868180","231577920","359464140","400756300","313293420","163194060","50908793","7199206"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","2","15","88","465","2314","11075","51580","235445","1058286","4699175","20660832","90099865","390225618","1680207115"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["2"]],"recurrence":{"denominator":{"coeffs":[["-2","-10"],["5","35"],["-1","-35"],["-2","10"]]},"terms":[{"coeff":{"coeffs":[["0","-8","-10","0"],["3","16","-25","-50"],["-5","-43","15","75"],["-2","11","0","-25"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-4","-9","-5","0"],["0","0","-20","-45","-25"],["0","-4","11","40","25"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":206276,"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-ada9039406d4b395"]},{"id":206277,"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-ada9039406d4b395"]},{"id":206278,"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-ada9039406d4b395"]}],"region_exact":{},"incoming_edges":[{"id":"e-ada9039406d4b395","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-caba3b00effe5aa9","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":"0f203b1885106b1264d7e09e698e5c2c415fc54324fa89f3cdef06cd1bb8908f","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-80f3c43c239c22ff","role":"sequence","name":"reversal of Bisection of Fibonacci triangle A037027: even-indexed members of column sequences of A037027 (not counting leading zeros).","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:  2;\n  2:  5,   10;\n  3:  9,   40,    39;\n  4: 14,  102,   213,   136;\n  5: 20,  210,   699,   940,    445;\n  6: 27,  380,  1782,  3768,   3720,   1398;\n  7: 35,  630,  3885, 11440,  17610,  13710,   4270;\n  8: 44,  980,  7602, 29140,  62010,  74838,  48055,  12776;\n  9: 54, 1452, 13734, 65628, 180610, 299778, 297129, 162272, 37629;\n","recurrence_sentence":"The row polynomials satisfy -(2*n^3+n^2-5*n+2-(10*n^3-35*n^2+35*n-10)*x)*P(n,x) = -(2*n^3+5*n^2-3*n-(11*n^3-43*n^2+16*n-8)*x-(15*n^2-25*n-10)*x^2+(25*n^3-75*n^2+50*n)*x^3)*P(n-1,x) - ((4*n^2+4)*x-(11*n^2-20*n-9)*x^2-(40*n^2-45*n-5)*x^3-(25*n^2-25*n)*x^4)*d/dx 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[Together[(-(2*n^3 + 5*n^2 - 3*n - (11*n^3 - 43*n^2 + 16*n - 8)*x - (15*n^2 - 25*n - 10)*x^2 + (25*n^3 - 75*n^2 + 50*n)*x^3)*P[n - 1, x] - ((4*n^2 + 4)*x - (11*n^2 - 20*n - 9)*x^2 - (40*n^2 - 45*n - 5)*x^3 - (25*n^2 - 25*n)*x^4)*D[P[n - 1, x], x])/(-2*n^3 - n^2 + 5*n - 2 + (10*n^3 - 35*n^2 + 35*n - 10)*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] = (-(2*n^3 + 5*n^2 - 3*n - (11*n^3 - 43*n^2 + 16*n - 8)*x - (15*n^2 - 25*n - 10)*x^2 + (25*n^3 - 75*n^2 + 50*n)*x^3)*v[i - 1] - ((4*n^2 + 4)*x - (11*n^2 - 20*n - 9)*x^2 - (40*n^2 - 45*n - 5)*x^3 - (25*n^2 - 25*n)*x^4)*deriv(v[i - 1], x))/(-2*n^3 - n^2 + 5*n - 2 + (10*n^3 - 35*n^2 + 35*n - 10)*x)); 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-fac4d4a000d744e8","steps":["reverse","D"],"edges":["e-0c6cc2be731e428d","e-ada9039406d4b395"],"text":"A060920 \u2192 reverse \u2192 D"},"headline":"A060920 \u2192 reverse \u2192 D","chain":null}
