{"id":"n-7ce3e84f14570a29","revision":0,"first_index":1,"fingerprint":"f6-fdb1bb4a756fa5c9","name":"one_plus_derivative of basis_map_A013609 of Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.","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"],["4","2"],["16","18","4"],["56","104","56","8"],["188","488","436","152","16"],["608","2040","2576","1496","384","32"],["1920","7912","12976","10696","4576","928","64"],["5952","29120","58880","63232","38464","12992","2176","128"],["18192","103104","248160","329344","261456","125632","35008","4992","256"],["54976","354336","989952","1566784","1539392","966176","382976","90752","11264","512"],["164608","1189280","3784512","6962752","8168448","6350752","3289280","1108352","228352","25088","1024"],["489088","3915648","13985152","29341312","40056192","37216384","23889536","10518912","3080192","561152","55296","2048"],["1443776","12688256","50268608","118487424","184624448","199760000","153318208","83688064","32018944","8286208","1352704","120832","4096"],["4238336","40566656","176572672","462048384","809462272","1000297600","894529792","584008576","277016576","93647872","21704704","3209216","262144","8192"],["12382208","128220288","608274176","1749954432","3405712896","4734629760","4838106880","3681575040","2089077760","875545600","265009152","55601152","7512064","565248","16384"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","6","38","224","1280","7136","39072","210944","1126144","5957120","31272448","163110912","846061568","4367572992","22452183040"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["6","8","-2","-2"],["-7","-14","0","2"],["0","6","2","0"],["1","0","0","0"]]},"terms":[{"coeff":{"coeffs":[["-12","-28","16","20","4"],["-10","34","16","0","0"],["12","-6","-4","0","0"],["-2","0","0","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","12","52","64","28","4"],["6","44","62","28","4","0"],["6","8","2","0","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":178057,"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-0e080700004e1dab"]},{"id":178058,"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-0e080700004e1dab"]},{"id":178059,"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-0e080700004e1dab"]}],"region_exact":{},"incoming_edges":[{"id":"e-0e080700004e1dab","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-7ce3e84f14570a29","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":"a3cfc934c732b11079d8b7de898ad2b6200f3bc226c6a9eb681a72fb60d0bbbe","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-726f071d833f296a","role":"sequence","name":"basis_map_A013609 of Triangle read by rows, T(n,k) = T(n-1,k) + T(n-2,k) + T(n-1,k-1) + T(n-2,k-1) and T(0,0) = 1.","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":[{"id":"l-cff27b79b0946475","kind":"fingerprint_related","node_a_id":"n-2f41a62f7aaca14e","node_b_id":"n-7ce3e84f14570a29","fingerprint":"f6-fdb1bb4a756fa5c9","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-fdb1bb4a756fa5c9"},"other_id":"n-2f41a62f7aaca14e","other":{"id":"n-2f41a62f7aaca14e","name":"one_plus_derivative of basis_map_A007318 of Riordan array (1, 2+2x).","first_index":0,"fingerprint":"f6-fdb1bb4a756fa5c9","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["4","2"],["16","18","4"]],"card_rows":[["1"],["4","2"],["16","18","4"],["56","104","56","8"]],"row_sums_preview":["1","6","38","224","1280","7136","39072","210944"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f5a001953ee28e8f","steps":["basis A007318","D+1"],"edges":["e-23c3ede392db8800","e-7fd99fa1601df75a"],"text":"A099040 \u2192 basis A007318 \u2192 D+1"},"headline":"A099040 \u2192 basis A007318 \u2192 D+1","chain":null}}],"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:     4,      2;\n   3:    16,     18,      4;\n   4:    56,    104,     56,       8;\n   5:   188,    488,    436,     152,      16;\n   6:   608,   2040,   2576,    1496,     384,     32;\n   7:  1920,   7912,  12976,   10696,    4576,    928,     64;\n   8:  5952,  29120,  58880,   63232,   38464,  12992,   2176,   128;\n   9: 18192, 103104, 248160,  329344,  261456, 125632,  35008,  4992,   256;\n  10: 54976, 354336, 989952, 1566784, 1539392, 966176, 382976, 90752, 11264, 512;\n","recurrence_sentence":"The row polynomials satisfy (n^3-7*n+6+(6*n^2-14*n+8)*x+(2*n^2-2)*x^2+(2*n-2)*x^3)*P(n,x) = -(2*n^3-12*n^2+10*n+12+(6*n^2-34*n+28)*x+(4*n^2-16*n-16)*x^2-20*x^3-4*x^4)*P(n-1,x) + (6*n^2+6*n+(8*n^2+44*n+12)*x+(2*n^2+62*n+52)*x^2+(28*n+64)*x^3+(4*n+28)*x^4+4*x^5)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.","mathematica":"P[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(2*n^3 - 12*n^2 + 10*n + 12 + (6*n^2 - 34*n + 28)*x + (4*n^2 - 16*n - 16)*x^2 - 20*x^3 - 4*x^4)*P[n - 1, x] + (6*n^2 + 6*n + (8*n^2 + 44*n + 12)*x + (2*n^2 + 62*n + 52)*x^2 + (28*n + 64)*x^3 + (4*n + 28)*x^4 + 4*x^5)*D[P[n - 1, x], x])/(n^3 - 7*n + 6 + (6*n^2 - 14*n + 8)*x + (2*n^2 - 2)*x^2 + (2*n - 2)*x^3)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(2*n^3 - 12*n^2 + 10*n + 12 + (6*n^2 - 34*n + 28)*x + (4*n^2 - 16*n - 16)*x^2 - 20*x^3 - 4*x^4)*v[i - 1] + (6*n^2 + 6*n + (8*n^2 + 44*n + 12)*x + (2*n^2 + 62*n + 52)*x^2 + (28*n + 64)*x^3 + (4*n + 28)*x^4 + 4*x^5)*deriv(v[i - 1], x))/(n^3 - 7*n + 6 + (6*n^2 - 14*n + 8)*x + (2*n^2 - 2)*x^2 + (2*n - 2)*x^3)); 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-c72653d9ab0ee8be","steps":["basis A013609","D+1"],"edges":["e-081205c1b1ff787d","e-0e080700004e1dab"],"text":"A063967 \u2192 basis A013609 \u2192 D+1"},"headline":"A063967 \u2192 basis A013609 \u2192 D+1","chain":null}