{"id":"n-9ca7cb1ef72efb17","revision":0,"first_index":0,"fingerprint":"f6-dd5843b5797216f1","name":"one_plus_derivative of basis_map_A007318 of One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.","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","2"],["32","22","3"],["160","176","52","4"],["768","1184","576","100","5"],["3584","7168","4992","1456","170","6"],["16384","40448","37120","15744","3136","266","7"],["73728","217088","248832","140800","41280","6048","392","8"],["327680","1122304","1548288","1104896","436480","95040","10752","552","9"],["1441792","5636096","9109504","7884800","3960320","1171456","198528","17952","750","10"],["6291456","27656192","51314688","52363264","32112640","12160512","2818816","384384","28512","990","11"],["27262976","133169152","279183360","328728576","238977024","111132672","33173504","6223360","700128","43472","1276","12"],["117440512","631242752","1476395008","1972633600","1662320640","920813568","339247104","82374656","12812800","1212640","64064","1612","13"],["503316480","2952790016","7625244672","11407458304","10950410240","7060193280","3111714816","936689664","189416448","24893440","2013440","91728","2002","14"],["2147483648","13656653824","38604374016","63963136000","68975329280","50836144128","26175012864","9462153216","2381414400","408478720","46052864","3224832","128128","2450","15"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","8","57","392","2633","17376","113105","728176","4646001","29421208","185131465","1158595512","7216558969","44764234544","276659588385"],"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"],["-1","-3","2"],["1","4","0"],["2","0","0"]]},"terms":[{"coeff":{"coeffs":[["0","0","1","-2"],["-8","-17","1","6"],["12","21","12","0"],["8","6","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-2","-6","-4"],["0","-8","-16","-8","0"],["0","-4","-4","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":179943,"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-7d26f4e1db113347"]},{"id":179944,"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-7d26f4e1db113347"]},{"id":179945,"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-7d26f4e1db113347"]},{"id":179946,"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-7d26f4e1db113347"]}],"region_exact":{},"incoming_edges":[{"id":"e-7d26f4e1db113347","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-9ca7cb1ef72efb17","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":"e0a8dad53c2885750d0c2df38c8d80b7dd637ed15e60060e36b092481f0aa6c0","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-3d55772447340781","role":"sequence","name":"basis_map_A007318 of One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.","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,       2;\n  2:      32,      22,       3;\n  3:     160,     176,      52,       4;\n  4:     768,    1184,     576,     100,       5;\n  5:    3584,    7168,    4992,    1456,     170,       6;\n  6:   16384,   40448,   37120,   15744,    3136,     266,      7;\n  7:   73728,  217088,  248832,  140800,   41280,    6048,    392,     8;\n  8:  327680, 1122304, 1548288, 1104896,  436480,   95040,  10752,   552,   9;\n  9: 1441792, 5636096, 9109504, 7884800, 3960320, 1171456, 198528, 17952, 750, 10;\n","recurrence_sentence":"The row polynomials satisfy (2*n^3+n^2-n+(4*n^2-3*n)*x+2*n*x^2)*P(n,x) = (8*n^3+12*n^2-8*n+(6*n^3+21*n^2-17*n)*x+(12*n^2+n+1)*x^2+(6*n-2)*x^3)*P(n-1,x) - ((4*n^2+8*n)*x+(4*n^2+16*n+2)*x^2+(8*n+6)*x^3+4*x^4)*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[((8*n^3 + 12*n^2 - 8*n + (6*n^3 + 21*n^2 - 17*n)*x + (12*n^2 + n + 1)*x^2 + (6*n - 2)*x^3)*P[n - 1, x] - ((4*n^2 + 8*n)*x + (4*n^2 + 16*n + 2)*x^2 + (8*n + 6)*x^3 + 4*x^4)*D[P[n - 1, x], x])/(2*n^3 + n^2 - n + (4*n^2 - 3*n)*x + 2*n*x^2)]];\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] = ((8*n^3 + 12*n^2 - 8*n + (6*n^3 + 21*n^2 - 17*n)*x + (12*n^2 + n + 1)*x^2 + (6*n - 2)*x^3)*v[i - 1] - ((4*n^2 + 8*n)*x + (4*n^2 + 16*n + 2)*x^2 + (8*n + 6)*x^3 + 4*x^4)*deriv(v[i - 1], x))/(2*n^3 + n^2 - n + (4*n^2 - 3*n)*x + 2*n*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-2245b35ec7cd4857","steps":["basis A007318","D+1"],"edges":["e-13cc0c6e042058d4","e-7d26f4e1db113347"],"text":"A091044 \u2192 basis A007318 \u2192 D+1"},"headline":"A091044 \u2192 basis A007318 \u2192 D+1","chain":null}