{"id":"n-f46badd442612bbe","revision":0,"first_index":0,"fingerprint":"f6-29fe05ab5e2e1f51","name":"theta_plus_one_squared of basis_map_A013611 of Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.","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"],["1","16"],["2","48","144"],["6","176","864","1024"],["24","800","5040","10240","6400"],["120","4384","32400","87040","96000","36864"],["720","28224","233856","752640","1120000","774144","200704"],["5040","209088","1891008","6931456","12544000","11870208","5619712","1048576"],["40320","1753344","17009856","68898816","143673600","167215104","109584384","37748736","5308416"],["362880","16425216","168868800","741048320","1723680000","2332495872","1896652800","912261120","238878720","26214400"],["3628800","170058240","1836514944","8611328000","21868352000","33253355520","31665672192","19031654400","7007109120","1441792000","126877696"],["39916800","1928701440","21732188544","107784269824","294372672000","491748618240","529368440832","374785179648","173425950720","50462720000","8373927936","603979776"],["479001600","23783086080","278144575488","1447951245312","4206123750400","7596570009600","9029719322624","7263102173184","3978461380608","1461977088000","344726700032","47110422528","2835349504"],["6227020800","316844144640","3829927256064","20801283170304","63729304038400","122982682927104","158745454579712","141596004712448","88489452699648","38652425011200","11557416206336","2253448544256","258016804864","13153337344"],["87178291200","4535450357760","56470578886656","318453002649600","1022218276352000","2088838352240640","2892008748941312","2811708073574400","1955682111651840","978118901760000","348881563942912","86565401395200","14190924267520","1381100421120","60397977600"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","17","194","2070","22504","256808","3110288","40119088","551232576","8056888128","125016342912","2054626565760","35680984104960","653231640453120","12579200062709760"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one_squared","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["1","16"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-1","16","144"],["2","-16","0"],["-1","0","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","-16","64"],["0","0","16","0"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-1","16"],["2","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-16"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":183461,"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-63bed6b856ab2af3"]},{"id":183462,"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-63bed6b856ab2af3"]},{"id":183463,"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-63bed6b856ab2af3"]},{"id":183464,"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-63bed6b856ab2af3"]}],"region_exact":{},"incoming_edges":[{"id":"e-63bed6b856ab2af3","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"2\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-f46badd442612bbe","premise_count":1,"theorem":"Laguerre's theorem on multiplier sequences: if f is a real polynomial whose roots are all real and <= 0, then (f(k))_(k >= 0) is a multiplier sequence, so x^k -> f(k) x^k maps every real-rooted polynomial to a real-rooted polynomial or zero (Laguerre; see T. Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"certificate_hash":"fe3d6cc01b1d87e02085de9c42af89f3b240e6f57b009328960e39419834c43b","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-126effb6ce513816","role":"sequence","name":"basis_map_A013611 of Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.","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,       16;\n  2:      2,       48,       144;\n  3:      6,      176,       864,      1024;\n  4:     24,      800,      5040,     10240,       6400;\n  5:    120,     4384,     32400,     87040,      96000,      36864;\n  6:    720,    28224,    233856,    752640,    1120000,     774144,     200704;\n  7:   5040,   209088,   1891008,   6931456,   12544000,   11870208,    5619712,   1048576;\n  8:  40320,  1753344,  17009856,  68898816,  143673600,  167215104,  109584384,  37748736,   5308416;\n  9: 362880, 16425216, 168868800, 741048320, 1723680000, 2332495872, 1896652800, 912261120, 238878720, 26214400;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^2-2*n+1+(16*n-16)*x-144*x^2)*P(n-2,x) + ((16*n-16)*x^2+64*x^3)*d/dx P(n-2,x) + (2*n-1+16*x)*P(n-1,x) - 16*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 16*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 16*x;\nP[n_, x_] := P[n, x] = Expand[-(n^2 - 2*n + 1 + (16*n - 16)*x - 144*x^2)*P[n - 2, x] + ((16*n - 16)*x^2 + 64*x^3)*D[P[n - 2, x], x] + (2*n - 1 + 16*x)*P[n - 1, x] - 16*x^2*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 16*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 - 2*n + 1 + (16*n - 16)*x - 144*x^2)*v[i - 2] + ((16*n - 16)*x^2 + 64*x^3)*deriv(v[i - 2], x) + (2*n - 1 + 16*x)*v[i - 1] - 16*x^2*deriv(v[i - 1], 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-b9fbfb57d9269e2f","steps":["basis A013611","(\u03b8+1)\u00b2"],"edges":["e-0a0b77ef6a10db4c","e-63bed6b856ab2af3"],"text":"A132393 \u2192 basis A013611 \u2192 (\u03b8+1)\u00b2"},"headline":"A132393 \u2192 basis A013611 \u2192 (\u03b8+1)\u00b2","chain":null}