{"id":"n-68a09e7004d283fc","revision":1,"first_index":0,"fingerprint":"f6-55c35b9641c5f83c","name":"theta_plus_one of Triangle of Bessel numbers read by rows: T(n,k) is the number of k-matchings of the complete graph K(n).","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","2"],["1","6"],["1","12","9"],["1","20","45"],["1","30","135","60"],["1","42","315","420"],["1","56","630","1680","525"],["1","72","1134","5040","4725"],["1","90","1890","12600","23625","5670"],["1","110","2970","27720","86625","62370"],["1","132","4455","55440","259875","374220","72765"],["1","156","6435","102960","675675","1621620","945945"],["1","182","9009","180180","1576575","5675670","6621615","1081080"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","1","3","7","22","66","226","778","2892","10972","43876","179796","766888","3352792","15144312"],"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":[["2","0"],["0","1"],["0","1"]]},"terms":[{"coeff":{"coeffs":[["2","-4","-2"],["0","5","-1"],["0","1","2"],["0","0","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-4","-4"],["0","0","0","-6"],["0","0","0","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":161455,"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-0e3cf0b8dd3f845e"]},{"id":161456,"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-0e3cf0b8dd3f845e"]},{"id":161457,"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-0e3cf0b8dd3f845e"]}],"region_exact":{},"incoming_edges":[{"id":"e-0e3cf0b8dd3f845e","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-68a09e7004d283fc","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":"65c81a7d5c4f70ae95ac460fa8bfc7aecef4157c782ab546b55067e594629f69","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-97b51734d39ede6c","role":"sequence","name":"Triangle of Bessel numbers read by rows: T(n,k) is the number of k-matchings of the complete graph 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;\n  2: 1,  2;\n  3: 1,  6;\n  4: 1, 12,    9;\n  5: 1, 20,   45;\n  6: 1, 30,  135,   60;\n  7: 1, 42,  315,  420;\n  8: 1, 56,  630, 1680,  525;\n  9: 1, 72, 1134, 5040, 4725;\n","recurrence_sentence":"The row polynomials satisfy (2+(n^2+n)*x)*P(n,x) = (2+(n^2+5*n-4)*x+(n^3+2*n^2-n-2)*x^2)*P(n-1,x) - (4*x^2+(2*n^2+6*n+4)*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[((2 + (n^2 + 5*n - 4)*x + (n^3 + 2*n^2 - n - 2)*x^2)*P[n - 1, x] - (4*x^2 + (2*n^2 + 6*n + 4)*x^3)*D[P[n - 1, x], x])/(2 + (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] = ((2 + (n^2 + 5*n - 4)*x + (n^3 + 2*n^2 - n - 2)*x^2)*v[i - 1] - (4*x^2 + (2*n^2 + 6*n + 4)*x^3)*deriv(v[i - 1], x))/(2 + (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-97b51734d39ede6c","steps":["\u03b8+1"],"edges":["e-0e3cf0b8dd3f845e"],"text":"A100861 \u2192 \u03b8+1"},"headline":"A100861 \u2192 \u03b8+1","chain":null}