{"id":"n-62d4fc6a50b318be","revision":0,"first_index":0,"fingerprint":"f6-29515cc327d9f4a5","name":"one_plus_derivative of basis_map_A013611 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"],["6","4"],["16","12"],["58","144","48"],["186","640","240"],["676","3480","4320","960"],["2416","15624","26880","6720"],["9276","75712","194880","161280","26880"],["36028","347904","1124928","1290240","241920"],["147016","1669680","6814080","11692800","7257600","967680"],["612096","7926160","38269440","82494720","70963200","10644480"],["2647096","38812224","220397760","599639040","771724800","383201280","42577920"],["11705656","190973952","1236480960","3980021760","6434588160","4428103680","553512960"],["53405808","963542944","7063824768","26741594880","54567152640","56181565440","23247544320","2214051840"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","1","10","28","250","1066","9436","51640","468028","3041020","28548856","210910096","2059000120","16835387128","171032682640"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["2","-64","32"],["-16","32","0"],["8","0","0"]]},"terms":[{"coeff":{"coeffs":[["-8","-8","128","-128"],["10","8","-224","128"],["-16","-64","128","0"],["8","32","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-1","-8","-32","-128","-256"],["0","-16","-128","-256","0"],["-4","-32","-64","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":176940,"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-2f3357677b22c345"]},{"id":176941,"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-2f3357677b22c345"]},{"id":176942,"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-2f3357677b22c345"]}],"region_exact":{},"incoming_edges":[{"id":"e-2f3357677b22c345","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-62d4fc6a50b318be","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":"3704c280f4bfa8d7074d5b02847206169a73c9a50994f870d3c7819c849a38f4","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-378328f3fde82415","role":"sequence","name":"basis_map_A013611 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":[],"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:     6,      4;\n  3:    16,     12;\n  4:    58,    144,      48;\n  5:   186,    640,     240;\n  6:   676,   3480,    4320,     960;\n  7:  2416,  15624,   26880,    6720;\n  8:  9276,  75712,  194880,  161280,  26880;\n  9: 36028, 347904, 1124928, 1290240, 241920;\n","recurrence_sentence":"The row polynomials satisfy (8*n^2-16*n+2+(32*n-64)*x+32*x^2)*P(n,x) = (8*n^3-16*n^2+10*n-8+(32*n^3-64*n^2+8*n-8)*x+(128*n^2-224*n+128)*x^2+(128*n-128)*x^3)*P(n-1,x) - (4*n^2+1+(32*n^2+16*n+8)*x+(64*n^2+128*n+32)*x^2+(256*n+128)*x^3+256*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[((8*n^3 - 16*n^2 + 10*n - 8 + (32*n^3 - 64*n^2 + 8*n - 8)*x + (128*n^2 - 224*n + 128)*x^2 + (128*n - 128)*x^3)*P[n - 1, x] - (4*n^2 + 1 + (32*n^2 + 16*n + 8)*x + (64*n^2 + 128*n + 32)*x^2 + (256*n + 128)*x^3 + 256*x^4)*D[P[n - 1, x], x])/(8*n^2 - 16*n + 2 + (32*n - 64)*x + 32*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] = ((8*n^3 - 16*n^2 + 10*n - 8 + (32*n^3 - 64*n^2 + 8*n - 8)*x + (128*n^2 - 224*n + 128)*x^2 + (128*n - 128)*x^3)*v[i - 1] - (4*n^2 + 1 + (32*n^2 + 16*n + 8)*x + (64*n^2 + 128*n + 32)*x^2 + (256*n + 128)*x^3 + 256*x^4)*deriv(v[i - 1], x))/(8*n^2 - 16*n + 2 + (32*n - 64)*x + 32*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":["basis A013611","D+1"],"edges":["e-ad55a1201a8e5f74","e-2f3357677b22c345"],"text":"A100861 \u2192 basis A013611 \u2192 D+1"},"headline":"A100861 \u2192 basis A013611 \u2192 D+1","chain":null}