{"id":"n-bb38c4896765a535","revision":0,"first_index":0,"fingerprint":"f6-73cb17ab6470879f","name":"one_plus_derivative of derivative 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":-2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[[],[],["1"],["3"],["12","6"],["40","30"],["105","180","45"],["231","840","315"],["448","2940","2520","420"],["792","8316","15120","3780"],["1305","20160","66150","37800","4725"],["2035","43560","228690","277200","51975"],["3036","86130","665280","1455300","623700","62370"],["4368","158730","1698840","5945940","5405400","810810"],["6097","276276","3918915","20180160","33108075","11351340","945945"]],"degrees":[-1,-1,0,0,1,1,2,2,3,3,4,4,5,5,6],"row_sums":["0","0","1","3","18","70","330","1386","6328","28008","130140","603460","2895816","14024088","69786808"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["0"],["1"]],"recurrence":{"denominator":{"coeffs":[["-28","36","-8"],["28","-26","4"],["-9","4","0"],["1","0","0"]]},"terms":[{"coeff":{"coeffs":[["0","0","0","0"],["8","-54","46","-12"],["-5","37","-26","4"],["1","-10","4","0"],["0","1","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","0","0","0"],["0","0","-16","20","-8"],["0","0","10","-8","0"],["0","0","-2","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":173392,"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-eaeb5b4376054dca"]},{"id":173393,"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-eaeb5b4376054dca"]},{"id":173394,"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-eaeb5b4376054dca"]}],"region_exact":{},"incoming_edges":[{"id":"e-eaeb5b4376054dca","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-bb38c4896765a535","premise_count":1,"theorem":"Laguerre's theorem on differential operators: if phi is a real polynomial with only real roots, then phi(D) maps every real-rooted polynomial to a real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"d0c53319bbd0b0ffeeec057174fad517431060235e387da9b5f4667e2da95da8","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-4c77cf1836e24da4","role":"sequence","name":"derivative of 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":[{"id":"l-d497d4038df6e0bb","kind":"fingerprint_related","node_a_id":"n-dcfa39142f813dd0","node_b_id":"n-bb38c4896765a535","fingerprint":"f6-73cb17ab6470879f","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-73cb17ab6470879f"},"other_id":"n-dcfa39142f813dd0","other":{"id":"n-dcfa39142f813dd0","name":"one_plus_derivative of Triangle of Bessel numbers read by rows: T(n,k) is the number of k-matchings of the complete graph K(n).","first_index":0,"fingerprint":"f6-73cb17ab6470879f","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["2","1"]],"card_rows":[["1"],["1"],["2","1"],["4","3"]],"row_sums_preview":["1","1","3","7","22","66","226","778"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-97b51734d39ede6c","steps":["D+1"],"edges":["e-11861e18b0a26fda"],"text":"A100861 \u2192 D+1"},"headline":"A100861 \u2192 D+1","chain":null}}],"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:   0;\n  1:   0;\n  2:   1;\n  3:   3;\n  4:  12,    6;\n  5:  40,   30;\n  6: 105,  180,    45;\n  7: 231,  840,   315;\n  8: 448, 2940,  2520,  420;\n  9: 792, 8316, 15120, 3780;\n","recurrence_sentence":"The row polynomials satisfy (n^3-9*n^2+28*n-28+(4*n^2-26*n+36)*x+(4*n-8)*x^2)*P(n,x) = (n^3-5*n^2+8*n+(n^4-10*n^3+37*n^2-54*n)*x+(4*n^3-26*n^2+46*n)*x^2+(4*n^2-12*n)*x^3)*P(n-1,x) - ((2*n^3-10*n^2+16*n)*x^2+(8*n^2-20*n)*x^3+8*n*x^4)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 0;\nP[2, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((n^3 - 5*n^2 + 8*n + (n^4 - 10*n^3 + 37*n^2 - 54*n)*x + (4*n^3 - 26*n^2 + 46*n)*x^2 + (4*n^2 - 12*n)*x^3)*P[n - 1, x] - ((2*n^3 - 10*n^2 + 16*n)*x^2 + (8*n^2 - 20*n)*x^3 + 8*n*x^4)*D[P[n - 1, x], x])/(n^3 - 9*n^2 + 28*n - 28 + (4*n^2 - 26*n + 36)*x + (4*n - 8)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 0; v[3] = 1; for(n = 3, N, my(i = n + 1); v[i] = ((n^3 - 5*n^2 + 8*n + (n^4 - 10*n^3 + 37*n^2 - 54*n)*x + (4*n^3 - 26*n^2 + 46*n)*x^2 + (4*n^2 - 12*n)*x^3)*v[i - 1] - ((2*n^3 - 10*n^2 + 16*n)*x^2 + (8*n^2 - 20*n)*x^3 + 8*n*x^4)*deriv(v[i - 1], x))/(n^3 - 9*n^2 + 28*n - 28 + (4*n^2 - 26*n + 36)*x + (4*n - 8)*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-97b51734d39ede6c","steps":["D","D+1"],"edges":["e-409924954694eb46","e-eaeb5b4376054dca"],"text":"A100861 \u2192 D \u2192 D+1"},"headline":"A100861 \u2192 D \u2192 D+1","chain":null}
