{"id":"n-8bc6801b2691aa61","revision":0,"first_index":1,"fingerprint":"f6-e80bd10399b19c76","name":"one_plus_derivative of basis_map_A007318 of Triangle read by rows: T(n,k) are the coefficients of Charlier polynomials: A046716 transposed, for 0 <= k <= n.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["3","1"],["10","7","1"],["39","41","12","1"],["181","246","107","18","1"],["995","1609","905","225","25","1"],["6386","11697","7879","2535","415","33","1"],["47075","94737","72933","28224","5985","700","42","1"],["392673","851204","726662","323288","82719","12544","1106","52","1"],["3659731","8429889","7818426","3881282","1146537","210315","24066","1662","63","1"],["37709394","91400995","90770901","49249650","16307360","3464685","480543","43110","2400","75","1"],["425779519","1078067737","1134245398","662618231","240868925","57562637","9270954","1009833","73095","3355","88","1"],["5228835525","13752523562","15205174665","9457819162","3717367126","978678690","177791669","22535766","1983498","118470","4565","102","1"],["69396404467","188754126729","217926898787","143093990819","60117294952","17170237942","3445936923","493798019","50666616","3684538","184899","6071","117","1"],["989898002666","2774395106077","3328132597763","2291313324937","1019819086923","312313211210","68182178064","10818764957","1257383699","106760654","6530524","279461","7917","133","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","18","93","553","3760","28946","249697","2390249","25171972","289429114","3609499773","48542832801","700453230880","10796243234986"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["3","1"],["10","7","1"],["39","41","12","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-12"],["7"],["-1"]]},"deriv_order":0,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["9","-6"],["-9","2"],["2","0"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["4","2","-1"],["0","-2","0"],["-1","0","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-1","2"],["2","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":180154,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-c8ee2257a1a885ef"]},{"id":180155,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-c8ee2257a1a885ef"]},{"id":180156,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-c8ee2257a1a885ef"]},{"id":180157,"position":3,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2","region_exact":null,"edges":["e-c8ee2257a1a885ef"]}],"region_exact":{},"incoming_edges":[{"id":"e-c8ee2257a1a885ef","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-8bc6801b2691aa61","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":"9c82316d24e62168342bee2e95c2348fb83250d8990ca7563cdf672429f3ae10","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-8addbc07b5fd134f","role":"sequence","name":"basis_map_A007318 of Triangle read by rows: T(n,k) are the coefficients of Charlier polynomials: A046716 transposed, for 0 <= k <= n.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:       1;\n   2:       3,       1;\n   3:      10,       7,       1;\n   4:      39,      41,      12,       1;\n   5:     181,     246,     107,      18,       1;\n   6:     995,    1609,     905,     225,      25,      1;\n   7:    6386,   11697,    7879,    2535,     415,     33,     1;\n   8:   47075,   94737,   72933,   28224,    5985,    700,    42,    1;\n   9:  392673,  851204,  726662,  323288,   82719,  12544,  1106,   52,  1;\n  10: 3659731, 8429889, 7818426, 3881282, 1146537, 210315, 24066, 1662, 63, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^2-7*n+12)*P(n-4,x) + (2*n^2-9*n+9+(2*n-6)*x)*P(n-3,x) - (n^2-4+(2*n-2)*x+x^2)*P(n-2,x) + (2*n-1+2*x)*P(n-1,x) for n >= 5, with P(1,x) = 1, P(2,x) = 3 + x, P(3,x) = 10 + 7*x + x^2, P(4,x) = 39 + 41*x + 12*x^2 + x^3.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 3 + x;\nP[3, x_] := 10 + 7*x + x^2;\nP[4, x_] := 39 + 41*x + 12*x^2 + x^3;\nP[n_, x_] := P[n, x] = Expand[-(n^2 - 7*n + 12)*P[n - 4, x] + (2*n^2 - 9*n + 9 + (2*n - 6)*x)*P[n - 3, x] - (n^2 - 4 + (2*n - 2)*x + x^2)*P[n - 2, x] + (2*n - 1 + 2*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 3 + x; v[3] = 10 + 7*x + x^2; v[4] = 39 + 41*x + 12*x^2 + x^3; for(n = 5, N, my(i = n); v[i] = -(n^2 - 7*n + 12)*v[i - 4] + (2*n^2 - 9*n + 9 + (2*n - 6)*x)*v[i - 3] - (n^2 - 4 + (2*n - 2)*x + x^2)*v[i - 2] + (2*n - 1 + 2*x)*v[i - 1]); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-3272d5bc697b9cb4","steps":["x \u21a6 x+1","D+1"],"edges":["e-77ff853dcb30433c","e-c8ee2257a1a885ef"],"text":"A094816 \u2192 x \u21a6 x+1 \u2192 D+1"},"headline":"A094816 \u2192 x \u21a6 x+1 \u2192 D+1","chain":null}