{"id":"n-d20569fbd513a05a","revision":0,"first_index":0,"fingerprint":"f6-db59e965f0bc5830","name":"derivative of reversal of Triangular array formed by the Mersenne 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":[[],["3"],["6","14"],["9","46","45"],["12","96","216","124"],["15","164","594","804","315"],["18","250","1260","2796","2610","762"],["21","354","2295","7180","11115","7746","1785"],["24","476","3780","15360","33840","39372","21588","4088"],["27","616","5796","29064","83790","139656","128268","57448","9207"],["30","774","8424","50344","180180","395220","522312","392232","147582","20470"],["33","950","11745","81576","349650","956676","1670970","1810600","1141533","368790","45045"],["36","1144","15840","125460","627480","2064384","4514832","6485720","5909220","3193320","901296","98292"],["39","1356","20790","185020","1058805","4078584","10761156","19414008","23501745","18364220","8648838","2162892","212979"],["42","1586","26676","263604","1699830","7513902","23279256","50804376","77414454","80489070","54806004","22804212","5112042","458738"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","3","20","100","448","1892","7696","30496","118528","453872","1717568","6437568","23937024","88410432","324673792"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["3"]],"recurrence":{"denominator":{"coeffs":[["6","4"],["-18","-6"],["12","2"]]},"terms":[{"coeff":{"coeffs":[["0","-8","6","2"],["-6","-66","-33","-3"],["12","56","15","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","1","6","1"],["0","-6","-37","-12","-1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":174513,"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-5c6950af578b55d9"]},{"id":174514,"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-5c6950af578b55d9"]},{"id":174515,"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-5c6950af578b55d9"]}],"region_exact":{},"incoming_edges":[{"id":"e-5c6950af578b55d9","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-d20569fbd513a05a","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":"5c1d4947b2a703d13a9416066bb0091238ed03510b74d1c4403337135576d599","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-369f42b9099fbd84","role":"sequence","name":"reversal of Triangular array formed by the Mersenne 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:  3;\n  2:  6,  14;\n  3:  9,  46,   45;\n  4: 12,  96,  216,   124;\n  5: 15, 164,  594,   804,   315;\n  6: 18, 250, 1260,  2796,  2610,    762;\n  7: 21, 354, 2295,  7180, 11115,   7746,   1785;\n  8: 24, 476, 3780, 15360, 33840,  39372,  21588,  4088;\n  9: 27, 616, 5796, 29064, 83790, 139656, 128268, 57448, 9207;\n","recurrence_sentence":"The row polynomials satisfy (12*n^2-18*n+6+(2*n^2-6*n+4)*x)*P(n,x) = (12*n^2-6*n+(56*n^2-66*n-8)*x+(15*n^2-33*n+6)*x^2+(n^2-3*n+2)*x^3)*P(n-1,x) - (6*n*x+(37*n-1)*x^2+(12*n-6)*x^3+(n-1)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 3;\nP[n_, x_] := P[n, x] = Expand[Together[((12*n^2 - 6*n + (56*n^2 - 66*n - 8)*x + (15*n^2 - 33*n + 6)*x^2 + (n^2 - 3*n + 2)*x^3)*P[n - 1, x] - (6*n*x + (37*n - 1)*x^2 + (12*n - 6)*x^3 + (n - 1)*x^4)*D[P[n - 1, x], x])/(12*n^2 - 18*n + 6 + (2*n^2 - 6*n + 4)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 3; for(n = 2, N, my(i = n + 1); v[i] = ((12*n^2 - 6*n + (56*n^2 - 66*n - 8)*x + (15*n^2 - 33*n + 6)*x^2 + (n^2 - 3*n + 2)*x^3)*v[i - 1] - (6*n*x + (37*n - 1)*x^2 + (12*n - 6)*x^3 + (n - 1)*x^4)*deriv(v[i - 1], x))/(12*n^2 - 18*n + 6 + (2*n^2 - 6*n + 4)*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-9b5c7e97e5191da6","steps":["reverse","D"],"edges":["e-5faf1c18373ba45e","e-5c6950af578b55d9"],"text":"A110441 \u2192 reverse \u2192 D"},"headline":"A110441 \u2192 reverse \u2192 D","chain":null}