{"id":"n-772e0ec3bbbc91e6","revision":0,"first_index":0,"fingerprint":"f6-b9815478474a59b6","name":"one_plus_derivative of basis_map_A013609 of Triangle T(n,k), read by rows given by [1,0,1,0,0,0,0,0,0,...] DELTA [0,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"irregular","degree_slope":null,"degree_intercept":null,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1"],["4","2"],["15","18","4"],["51","102","56","8"],["164","466","432","152","16"],["509","1884","2508","1488","384","32"],["1541","7044","12324","10512","4560","928","64"],["4580","24946","54288","61000","38000","12960","2176","128"],["13419","84894","221408","310264","254640","124512","34944","4992","256"],["38871","280218","852796","1435856","1470768","946848","380352","90624","11264","512"],["111556","902882","3142944","6189584","7625632","6126624","3237248","1102336","228096","25088","1024"],["317689","2852760","11184072","25246656","36423120","35206976","23206400","10384128","3066624","560640","55296","2048"],["898825","8868840","38678520","98520000","163110000","184715200","146469120","81711360","31680000","8256000","1351680","120832","4096"],["2528836","27199010","130634208","370774384","693430880","901705824","837778560","562086912","271524096","92815360","21638144","3207168","262144","8192"]],"degrees":[0,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["1","1","6","37","217","1230","6805","36973","198078","1049329","5508109","28693014","148506409","764384473","3915593718"],"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":[["-66","24","24"],["116","-16","-16"],["-72","0","0"],["16","0","0"]]},"terms":[{"coeff":{"coeffs":[["-4","68","64","16"],["30","-44","-56","-16"],["-56","-32","0","0"],["24","16","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-25","-120","-184","-96","-16"],["-40","-96","-32","0","0"],["20","48","16","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":175301,"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-c8223ad93239469f"]},{"id":175302,"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-c8223ad93239469f"]},{"id":175303,"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-c8223ad93239469f"]},{"id":175304,"position":3,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2","region_exact":null,"edges":["e-c8223ad93239469f"]}],"region_exact":{},"incoming_edges":[{"id":"e-c8223ad93239469f","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-772e0ec3bbbc91e6","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":"c32c4acb45ed30597a5b3df40c4aa2c4c1450c01c7247ff28fceaead3c44b65b","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-1852b727a5d6b55d","role":"sequence","name":"basis_map_A013609 of Triangle T(n,k), read by rows given by [1,0,1,0,0,0,0,0,0,...] DELTA [0,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.","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:     1;\n  2:     4,     2;\n  3:    15,    18,      4;\n  4:    51,   102,     56,      8;\n  5:   164,   466,    432,    152,     16;\n  6:   509,  1884,   2508,   1488,    384,     32;\n  7:  1541,  7044,  12324,  10512,   4560,    928,    64;\n  8:  4580, 24946,  54288,  61000,  38000,  12960,  2176,  128;\n  9: 13419, 84894, 221408, 310264, 254640, 124512, 34944, 4992, 256;\n","recurrence_sentence":"The row polynomials satisfy (16*n^3-72*n^2+116*n-66-(16*n-24)*x-(16*n-24)*x^2)*P(n,x) = (24*n^3-56*n^2+30*n-4+(16*n^3-32*n^2-44*n+68)*x-(56*n-64)*x^2-(16*n-16)*x^3)*P(n-1,x) + (20*n^2-40*n-25+(48*n^2-96*n-120)*x+(16*n^2-32*n-184)*x^2-96*x^3-16*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[((24*n^3 - 56*n^2 + 30*n - 4 + (16*n^3 - 32*n^2 - 44*n + 68)*x - (56*n - 64)*x^2 - (16*n - 16)*x^3)*P[n - 1, x] + (20*n^2 - 40*n - 25 + (48*n^2 - 96*n - 120)*x + (16*n^2 - 32*n - 184)*x^2 - 96*x^3 - 16*x^4)*D[P[n - 1, x], x])/(16*n^3 - 72*n^2 + 116*n - 66 - (16*n - 24)*x - (16*n - 24)*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] = ((24*n^3 - 56*n^2 + 30*n - 4 + (16*n^3 - 32*n^2 - 44*n + 68)*x - (56*n - 64)*x^2 - (16*n - 16)*x^3)*v[i - 1] + (20*n^2 - 40*n - 25 + (48*n^2 - 96*n - 120)*x + (16*n^2 - 32*n - 184)*x^2 - 96*x^3 - 16*x^4)*deriv(v[i - 1], x))/(16*n^3 - 72*n^2 + 116*n - 66 - (16*n - 24)*x - (16*n - 24)*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-0eba401b117f16a9","steps":["basis A013609","D+1"],"edges":["e-354781356c20418d","e-c8223ad93239469f"],"text":"A085478 \u2192 basis A013609 \u2192 D+1"},"headline":"A085478 \u2192 basis A013609 \u2192 D+1","chain":null}