{"id":"n-e084df69abd1b0bd","revision":0,"first_index":0,"fingerprint":"f6-8f36181a92323739","name":"one_plus_derivative of basis_map_A013610 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":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["6","3"],["31","39","9"],["138","303","180","27"],["574","1833","1899","729","81"],["2286","9648","14634","9747","2754","243"],["8839","46530","93879","91557","44631","9963","729"],["33450","211341","534708","695169","495234","189783","34992","2187"],["124537","918939","2802663","4581063","4350996","2430729","766179","120285","6561"],["457752","3865389","13817322","27293247","32623398","24279102","11128914","2976507","406782","19683"],["1665214","15841419","64994382","150811821","218127087","204319260","124615989","48356757","11225871","1358127","59049"],["6006348","63574488","294573132","786091446","1337416758","1516719249","1163865996","600158727","201724506","41353983","4487724","177147"],["21510541","250767972","1295491158","3911179662","7662916872","10230068907","9505022031","6162815097","2749072122","814462857","149453019","14703201","531441"],["76569774","974947935","5557328244","18735469506","41579522478","63975818682","69998041998","54942113619","30793121838","12092369148","3201518682","531618147","47829690","1594323"],["271141555","3744171975","23346037845","86961727710","215754998355","376347593304","474809296203","438900734709","297682273719","146761211286","51442248771","12306579237","1865889351","154649331","4782969"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","9","79","648","5116","39312","296128","2196864","16101952","116868096","841374976","6016149504","42767994880","302507864064","2130353336320"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0","0","0","0","0"],["33","369","-486","81","243"],["261","0","486","243","0"],["-81","243","0","0","0"]]},"terms":[{"coeff":{"coeffs":[["-363","-2151","-2862","1782","2673","729"],["1092","4842","5427","1134","-243","0"],["1197","2592","-729","-972","0","0"],["-648","-972","0","0","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-26","57","1773","6750","9396","4617","729"],["273","2268","6777","8667","4374","729","0"],["351","1863","2673","729","0","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":181240,"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-ba6d9b78d90f4cd3"]},{"id":181241,"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-ba6d9b78d90f4cd3"]},{"id":181242,"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-ba6d9b78d90f4cd3"]}],"region_exact":{},"incoming_edges":[{"id":"e-ba6d9b78d90f4cd3","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-e084df69abd1b0bd","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":"47aef04fdc53b98954f86655d076b746de9f92b3023b69bdf432421cf739994f","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-59c36108e7f99c2f","role":"sequence","name":"basis_map_A013610 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:      1;\n  1:      6,       3;\n  2:     31,      39,        9;\n  3:    138,     303,      180,       27;\n  4:    574,    1833,     1899,      729,       81;\n  5:   2286,    9648,    14634,     9747,     2754,      243;\n  6:   8839,   46530,    93879,    91557,    44631,     9963,      729;\n  7:  33450,  211341,   534708,   695169,   495234,   189783,    34992,    2187;\n  8: 124537,  918939,  2802663,  4581063,  4350996,  2430729,   766179,  120285,   6561;\n  9: 457752, 3865389, 13817322, 27293247, 32623398, 24279102, 11128914, 2976507, 406782, 19683;\n","recurrence_sentence":"The row polynomials satisfy -(81*n^3-261*n^2-33*n-(243*n^3+369*n)*x-(486*n^2-486*n)*x^2-(243*n^2+81*n)*x^3-243*n*x^4)*P(n,x) = -(648*n^3-1197*n^2-1092*n+363+(972*n^3-2592*n^2-4842*n+2151)*x+(729*n^2-5427*n+2862)*x^2+(972*n^2-1134*n-1782)*x^3+(243*n-2673)*x^4-729*x^5)*P(n-1,x) + (351*n^2+273*n-26+(1863*n^2+2268*n+57)*x+(2673*n^2+6777*n+1773)*x^2+(729*n^2+8667*n+6750)*x^3+(4374*n+9396)*x^4+(729*n+4617)*x^5+729*x^6)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(648*n^3 - 1197*n^2 - 1092*n + 363 + (972*n^3 - 2592*n^2 - 4842*n + 2151)*x + (729*n^2 - 5427*n + 2862)*x^2 + (972*n^2 - 1134*n - 1782)*x^3 + (243*n - 2673)*x^4 - 729*x^5)*P[n - 1, x] + (351*n^2 + 273*n - 26 + (1863*n^2 + 2268*n + 57)*x + (2673*n^2 + 6777*n + 1773)*x^2 + (729*n^2 + 8667*n + 6750)*x^3 + (4374*n + 9396)*x^4 + (729*n + 4617)*x^5 + 729*x^6)*D[P[n - 1, x], x])/(-81*n^3 + 261*n^2 + 33*n + (243*n^3 + 369*n)*x + (486*n^2 - 486*n)*x^2 + (243*n^2 + 81*n)*x^3 + 243*n*x^4)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(648*n^3 - 1197*n^2 - 1092*n + 363 + (972*n^3 - 2592*n^2 - 4842*n + 2151)*x + (729*n^2 - 5427*n + 2862)*x^2 + (972*n^2 - 1134*n - 1782)*x^3 + (243*n - 2673)*x^4 - 729*x^5)*v[i - 1] + (351*n^2 + 273*n - 26 + (1863*n^2 + 2268*n + 57)*x + (2673*n^2 + 6777*n + 1773)*x^2 + (729*n^2 + 8667*n + 6750)*x^3 + (4374*n + 9396)*x^4 + (729*n + 4617)*x^5 + 729*x^6)*deriv(v[i - 1], x))/(-81*n^3 + 261*n^2 + 33*n + (243*n^3 + 369*n)*x + (486*n^2 - 486*n)*x^2 + (243*n^2 + 81*n)*x^3 + 243*n*x^4)); 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 A013610","D+1"],"edges":["e-16250729b41a60db","e-ba6d9b78d90f4cd3"],"text":"A121574 \u2192 basis A013610 \u2192 D+1"},"headline":"A121574 \u2192 basis A013610 \u2192 D+1","chain":null}