{"id":"n-344833a32d1afc96","revision":1,"first_index":0,"fingerprint":"f6-4da4a97ce74eee20","name":"theta_plus_one of Pascal-(1,8,1) array.","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","2"],["1","20","3"],["1","38","57","4"],["1","56","354","112","5"],["1","74","894","1192","185","6"],["1","92","1677","6160","2795","276","7"],["1","110","2703","17932","22415","5406","385","8"],["1","128","3972","39424","106430","59136","9268","512","9"],["1","146","5484","73552","335030","402036","128716","14624","657","10"],["1","164","7239","123232","821210","1825800","1149694","246464","21717","820","11"],["1","182","9237","191380","1710770","6027540","7032130","2737232","430605","30790","1001","12"],["1","200","11478","280912","3182315","15964080","31105060","21285440","5728167","702280","42086","1200","13"],["1","218","13962","394744","5447255","36206250","105986860","121127840","54309375","10894510","1085546","55848","1417","14"],["1","236","16689","535792","8749805","73293180","297225565","526644800","382147155","122155300","19249571","1607376","72319","1652","15"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","24","100","528","2352","11008","48960","218880","960256","4196352","18170880","78303232","335523840","1431699456"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0","0","0"],["-1","2","17"],["0","-9","9"]]},"terms":[{"coeff":{"coeffs":[["0","34","292","34"],["-1","-34","-127","18"],["0","-9","-153","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","51","579","17"],["0","0","9","306","9"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":164860,"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-bf7cc7e5e6b812c4"]},{"id":164861,"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-bf7cc7e5e6b812c4"]},{"id":164862,"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-bf7cc7e5e6b812c4"]},{"id":164863,"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-bf7cc7e5e6b812c4"]}],"region_exact":{},"incoming_edges":[{"id":"e-bf7cc7e5e6b812c4","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-344833a32d1afc96","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":"72882678ba910f69c7437d2fbbc92519c30ab4ec2080334da37f74c4c1c41d03","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-1c472d1762258215","role":"sequence","name":"Pascal-(1,8,1) array.","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,   2;\n  2: 1,  20,    3;\n  3: 1,  38,   57,     4;\n  4: 1,  56,  354,   112,      5;\n  5: 1,  74,  894,  1192,    185,      6;\n  6: 1,  92, 1677,  6160,   2795,    276,      7;\n  7: 1, 110, 2703, 17932,  22415,   5406,    385,     8;\n  8: 1, 128, 3972, 39424, 106430,  59136,   9268,   512,   9;\n  9: 1, 146, 5484, 73552, 335030, 402036, 128716, 14624, 657, 10;\n","recurrence_sentence":"The row polynomials satisfy -(n+(9*n^2-2*n)*x-(9*n^2+17*n)*x^2)*P(n,x) = -(n+(9*n^2+34*n-34)*x+(153*n^2+127*n-292)*x^2-(18*n+34)*x^3)*P(n-1,x) + (x+(9*n+51)*x^2+(306*n+579)*x^3+(9*n+17)*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[(-(n + (9*n^2 + 34*n - 34)*x + (153*n^2 + 127*n - 292)*x^2 - (18*n + 34)*x^3)*P[n - 1, x] + (x + (9*n + 51)*x^2 + (306*n + 579)*x^3 + (9*n + 17)*x^4)*D[P[n - 1, x], x])/(-n - (9*n^2 - 2*n)*x + (9*n^2 + 17*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] = (-(n + (9*n^2 + 34*n - 34)*x + (153*n^2 + 127*n - 292)*x^2 - (18*n + 34)*x^3)*v[i - 1] + (x + (9*n + 51)*x^2 + (306*n + 579)*x^3 + (9*n + 17)*x^4)*deriv(v[i - 1], x))/(-n - (9*n^2 - 2*n)*x + (9*n^2 + 17*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-1c472d1762258215","steps":["\u03b8+1"],"edges":["e-bf7cc7e5e6b812c4"],"text":"A143683 \u2192 \u03b8+1"},"headline":"A143683 \u2192 \u03b8+1","chain":null}
