{"id":"n-fd271958fce1b7c5","revision":1,"first_index":0,"fingerprint":"f6-eb658cc66b4f4727","name":"theta_plus_one of Triangle read by rows: row n consists of the nonzero coefficients of the expansion of 2^n x^n in terms of Hermite polynomials with decreasing subscripts.","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"],["1"],["1","4"],["1","12"],["1","24","36"],["1","40","180"],["1","60","540","480"],["1","84","1260","3360"],["1","112","2520","13440","8400"],["1","144","4536","40320","75600"],["1","180","7560","100800","378000","181440"],["1","220","11880","221760","1386000","1995840"],["1","264","17820","443520","4158000","11975040","4656960"],["1","312","25740","823680","10810800","51891840","60540480"],["1","364","36036","1441440","25225200","181621440","423783360","138378240"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","1","5","13","61","221","1081","4705","24473","120601","667981","3615701","21251605","124092853","770486081"],"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":[["1","0"],["0","1"],["0","1"]]},"terms":[{"coeff":{"coeffs":[["1","-4","-4"],["0","5","-2"],["0","1","4"],["0","0","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-4","-8"],["0","0","0","-12"],["0","0","0","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":164016,"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-0f564e87cb0e9c13"]},{"id":164017,"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-0f564e87cb0e9c13"]},{"id":164018,"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-0f564e87cb0e9c13"]}],"region_exact":{},"incoming_edges":[{"id":"e-0f564e87cb0e9c13","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-fd271958fce1b7c5","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":"9f827e38048f2dc8b9c0dae67c0cef93c6d7193b12b7aeb5391915b20fc8661b","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-c914b69838ac70ec","role":"sequence","name":"Triangle read by rows: row n consists of the nonzero coefficients of the expansion of 2^n x^n in terms of Hermite polynomials with decreasing subscripts.","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: 1,   4;\n  3: 1,  12;\n  4: 1,  24,   36;\n  5: 1,  40,  180;\n  6: 1,  60,  540,   480;\n  7: 1,  84, 1260,  3360;\n  8: 1, 112, 2520, 13440,  8400;\n  9: 1, 144, 4536, 40320, 75600;\n","recurrence_sentence":"The row polynomials satisfy (1+(n^2+n)*x)*P(n,x) = (1+(n^2+5*n-4)*x+(2*n^3+4*n^2-2*n-4)*x^2)*P(n-1,x) - (4*x^2+(4*n^2+12*n+8)*x^3)*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[((1 + (n^2 + 5*n - 4)*x + (2*n^3 + 4*n^2 - 2*n - 4)*x^2)*P[n - 1, x] - (4*x^2 + (4*n^2 + 12*n + 8)*x^3)*D[P[n - 1, x], x])/(1 + (n^2 + n)*x)]];\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] = ((1 + (n^2 + 5*n - 4)*x + (2*n^3 + 4*n^2 - 2*n - 4)*x^2)*v[i - 1] - (4*x^2 + (4*n^2 + 12*n + 8)*x^3)*deriv(v[i - 1], x))/(1 + (n^2 + 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-c914b69838ac70ec","steps":["\u03b8+1"],"edges":["e-0f564e87cb0e9c13"],"text":"A059344 \u2192 \u03b8+1"},"headline":"A059344 \u2192 \u03b8+1","chain":null}