{"id":"n-d8c3e28cf4287787","revision":0,"first_index":0,"fingerprint":"f6-b4e1b99a45237f97","name":"one_plus_derivative of basis_map_A013610 of Triangle T(n,k)=Sum_{i=0..n} |stirling1(n,n-i)|*binomial(i,k), k=0..n-1.","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"],["48","69","18"],["471","1212","909","162"],["5499","20952","27801","14310","1944"],["74664","381267","743076","672435","265194","29160"],["1157742","7457265","19375776","25624485","17700444","5677452","524880"],["20203965","157646304","514803114","904367700","909872109","509962716","138122172","11022480"],["392116005","3601128888","14208796602","31290073092","41708719989","33966469044","16050337596","3766853808","264539520"],["8380608570","88665308667","410984491572","1091432441034","1820184760638","1960148018007","1343458578132","549493853940","113873870736","7142567040"],["195634652220","2345166123957","12504725020602","38954763778734","78226309569708","105237868488057","95239607889222","56426159724660","20363889924936","3780487041120","214277011200"],["4953324250035","66396980648556","400693290472575","1434269423514654","3374675643245085","5460240170497032","6165785421663741","4817920488331302","2516039178761556","813004228129608","136755584984160","7071141369600"],["135217535406975","2005179433060320","13519301594913819","54719454187299474","147799575103681449","279932195095408116","379818411423108993","369968165307644706","254582774506523196","118940377580640504","34810070914338048","5354540682975360","254561089305600"],["3959201745632700","64378484164728315","479841562101088008","2168222805609583581","6617275724342533770","14376663685780619913","22816293487071508044","26683355262461172363","22870926070342348446","14069040121996845588","5949521940740090184","1591857860940536832","225628118530208640","9927882482918400"],["123782088583777050","2190583749449839185","17892589185631363752","89325159657078444867","304159355128897911162","746195343137029295199","1357571279090489959116","1855845262396144928997","1908577438025247550008","1461465417824779204440","813237353984467676736","314232743926718105040","77457002739473913984","10181260634555969280","416971064282572800"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","9","135","2754","70506","2165796","77518044","3166000560","145249034544","7393764498336","413488889224416","25197804875867904","1661839824454306560","117926892208309814784","8958871542632830511616"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["6","3"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0","0"],["-1","-6","-9"],["4","18","18"],["-4","-12","-9"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-1","0"],["-2","-6"],["4","6"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":190497,"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-bfeb03c6c59c8733"]},{"id":190498,"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-bfeb03c6c59c8733"]},{"id":190499,"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-bfeb03c6c59c8733"]},{"id":190500,"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-bfeb03c6c59c8733"]}],"region_exact":{},"incoming_edges":[{"id":"e-bfeb03c6c59c8733","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-d8c3e28cf4287787","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":"06eee259faf649c7b7814fc73fe2f0f08a941e4d0f18be0e0f62c45a49f156cb","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-629fc911ba71fa1d","role":"sequence","name":"basis_map_A013610 of Triangle T(n,k)=Sum_{i=0..n} |stirling1(n,n-i)|*binomial(i,k), k=0..n-1.","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:         48,          69,           18;\n  3:        471,        1212,          909,           162;\n  4:       5499,       20952,        27801,         14310,          1944;\n  5:      74664,      381267,       743076,        672435,        265194,         29160;\n  6:    1157742,     7457265,     19375776,      25624485,      17700444,       5677452,        524880;\n  7:   20203965,   157646304,    514803114,     904367700,     909872109,     509962716,     138122172,     11022480;\n  8:  392116005,  3601128888,  14208796602,   31290073092,   41708719989,   33966469044,   16050337596,   3766853808,    264539520;\n  9: 8380608570, 88665308667, 410984491572, 1091432441034, 1820184760638, 1960148018007, 1343458578132, 549493853940, 113873870736, 7142567040;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = -(4*n^3-4*n^2+n+(12*n^3-18*n^2+6*n)*x+(9*n^3-18*n^2+9*n)*x^2)*P(n-2,x) + (4*n^2-2*n-1+(6*n^2-6*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 6 + 3*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 6 + 3*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(4*n^3 - 4*n^2 + n + (12*n^3 - 18*n^2 + 6*n)*x + (9*n^3 - 18*n^2 + 9*n)*x^2)*P[n - 2, x] + (4*n^2 - 2*n - 1 + (6*n^2 - 6*n)*x)*P[n - 1, x])/(n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 6 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (-(4*n^3 - 4*n^2 + n + (12*n^3 - 18*n^2 + 6*n)*x + (9*n^3 - 18*n^2 + 9*n)*x^2)*v[i - 2] + (4*n^2 - 2*n - 1 + (6*n^2 - 6*n)*x)*v[i - 1])/(n - 1)); 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-7fd16a32c438d522","steps":["basis A013610","D+1"],"edges":["e-07b12f78e33af42b","e-bfeb03c6c59c8733"],"text":"A059364 \u2192 basis A013610 \u2192 D+1"},"headline":"A059364 \u2192 basis A013610 \u2192 D+1","chain":null}