{"id":"n-9800635cf8538232","revision":0,"first_index":0,"fingerprint":"f6-1d7a93365d110f47","name":"theta_plus_one of veronese_section_2_0 of Cube of lower triangular normalized binomial matrix.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["3"],["9","2"],["27","18"],["81","108","3"],["243","540","45"],["729","2430","405","4"],["2187","10206","2835","84"],["6561","40824","17010","1008","5"],["19683","157464","91854","9072","135"],["59049","590490","459270","68040","2025","6"],["177147","2165130","2165130","449064","22275","198"],["531441","7794468","9743085","2694384","200475","3564","7"],["1594323","27634932","42220035","15011568","1563705","46332","273"],["4782969","96722262","177324147","78810732","10945935","486486","5733","8"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","3","11","45","192","828","3568","15312","65408","278208","1178880","4978944","20967424","88071168","369078272"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["3"]],"recurrence":{"denominator":{"coeffs":[["162","0"],["-162","-3"],["0","0"],["0","3"]]},"terms":[{"coeff":{"coeffs":[["486","108","-2"],["-486","-117","-1"],["0","0","2"],["0","9","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-972","144","-4"],["0","0","54","-6"],["0","0","18","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":176141,"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-947cde2baf7d6044"]},{"id":176142,"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-947cde2baf7d6044"]},{"id":176143,"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-947cde2baf7d6044"]}],"region_exact":{},"incoming_edges":[{"id":"e-947cde2baf7d6044","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-9800635cf8538232","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":"e69dd03ae83fd08701d27374e663fa9df48ad383d02b4f8cc2847349aa9a0fb9","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-081b3f700db2d097","role":"sequence","name":"veronese_section_2_0 of Cube of lower triangular normalized binomial matrix.","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:     3;\n  2:     9,      2;\n  3:    27,     18;\n  4:    81,    108,     3;\n  5:   243,    540,    45;\n  6:   729,   2430,   405,    4;\n  7:  2187,  10206,  2835,   84;\n  8:  6561,  40824, 17010, 1008,   5;\n  9: 19683, 157464, 91854, 9072, 135;\n","recurrence_sentence":"The row polynomials satisfy -(162*n-162-(3*n^3-3*n)*x)*P(n,x) = -(486*n-486-(9*n^3-117*n+108)*x-(n^3+2*n^2-n-2)*x^2)*P(n-1,x) - (972*x-(18*n^2+54*n+144)*x^2+(2*n^2+6*n+4)*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3;\nP[n_, x_] := P[n, x] = Expand[Together[(-(486*n - 486 - (9*n^3 - 117*n + 108)*x - (n^3 + 2*n^2 - n - 2)*x^2)*P[n - 1, x] - (972*x - (18*n^2 + 54*n + 144)*x^2 + (2*n^2 + 6*n + 4)*x^3)*D[P[n - 1, x], x])/(-162*n + 162 + (3*n^3 - 3*n)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3; for(n = 2, N, my(i = n + 1); v[i] = (-(486*n - 486 - (9*n^3 - 117*n + 108)*x - (n^3 + 2*n^2 - n - 2)*x^2)*v[i - 1] - (972*x - (18*n^2 + 54*n + 144)*x^2 + (2*n^2 + 6*n + 4)*x^3)*deriv(v[i - 1], x))/(-162*n + 162 + (3*n^3 - 3*n)*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-8d887bfc40d036b9","steps":["Veronese 2","\u03b8+1"],"edges":["e-5d982f8f22366510","e-947cde2baf7d6044"],"text":"A027465 \u2192 Veronese 2 \u2192 \u03b8+1"},"headline":"A027465 \u2192 Veronese 2 \u2192 \u03b8+1","chain":null}