{"id":"n-5d602378ea445bcb","revision":0,"first_index":0,"fingerprint":"f6-87f68aa23f424d68","name":"one_plus_derivative of basis_map_A013609 of A convolution triangle of numbers based on A001906 (even-indexed Fibonacci numbers).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["6","2"],["31","24","4"],["148","188","72","8"],["673","1208","756","192","16"],["2962","6918","6112","2528","480","32"],["12735","36752","42264","24640","7600","1152","64"],["53800","185208","263376","199760","86720","21312","2688","128"],["224193","897552","1523928","1429312","806640","278656","56896","6144","256"],["924062","4220682","8341344","9343328","6518624","2925216","838656","146432","13824","512"],["3774687","19378792","43730748","57038272","47523280","26353152","9807616","2402304","366336","30720","1024"],["15304124","87263796","221508056","330035480","320229120","212027648","97466880","30970368","6620160","896000","67584","2048"],["61656929","386677768","1090896300","1829102720","2027736160","1563452928","857918208","336574464","93277440","17684480","2151424","147456","4096"],["247054634","1690343950","5248319040","9784470080","12209396800","10756255680","6872429568","3217523712","1100487168","270364160","46047232","5087232","319488","8192"],["985270655","7304207520","24755711600","50813413760","70519026720","69936075008","51053778048","27826292736","11354610432","3441422336","759171072","117358592","11874304","688128","16384"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","8","59","416","2845","19032","125207","812992","5223577","33272680","210406931","1322391264","8267280373","51448106936","318878917295"],"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":[["0","0","0"],["2","-1","-1"],["0","0","0"],["1","0","0"]]},"terms":[{"coeff":{"coeffs":[["3","-3","-4","-1"],["7","-1","-4","-1"],["6","3","0","0"],["2","1","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-3","-13","-16","-7","-1"],["3","4","1","0","0"],["3","4","1","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":180095,"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-143d48ccab264415"]},{"id":180096,"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-143d48ccab264415"]},{"id":180097,"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-143d48ccab264415"]},{"id":180098,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-143d48ccab264415"]}],"region_exact":{},"incoming_edges":[{"id":"e-143d48ccab264415","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-5d602378ea445bcb","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":"3f164cd0a8d7e578b4103fe05bdff9ce0b49b19a5be815fadd3427b43f402da8","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-7d7dad3bd149bf81","role":"sequence","name":"basis_map_A013609 of A convolution triangle of numbers based on A001906 (even-indexed Fibonacci numbers).","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:      6,       2;\n  2:     31,      24,       4;\n  3:    148,     188,      72,       8;\n  4:    673,    1208,     756,     192,      16;\n  5:   2962,    6918,    6112,    2528,     480,      32;\n  6:  12735,   36752,   42264,   24640,    7600,    1152,     64;\n  7:  53800,  185208,  263376,  199760,   86720,   21312,   2688,    128;\n  8: 224193,  897552, 1523928, 1429312,  806640,  278656,  56896,   6144,   256;\n  9: 924062, 4220682, 8341344, 9343328, 6518624, 2925216, 838656, 146432, 13824, 512;\n","recurrence_sentence":"The row polynomials satisfy (n^3+2*n-n*x-n*x^2)*P(n,x) = (2*n^3+6*n^2+7*n+3+(n^3+3*n^2-n-3)*x-(4*n+4)*x^2-(n+1)*x^3)*P(n-1,x) + (3*n^2+3*n-3+(4*n^2+4*n-13)*x+(n^2+n-16)*x^2-7*x^3-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[((2*n^3 + 6*n^2 + 7*n + 3 + (n^3 + 3*n^2 - n - 3)*x - (4*n + 4)*x^2 - (n + 1)*x^3)*P[n - 1, x] + (3*n^2 + 3*n - 3 + (4*n^2 + 4*n - 13)*x + (n^2 + n - 16)*x^2 - 7*x^3 - x^4)*D[P[n - 1, x], x])/(n^3 + 2*n - n*x - n*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] = ((2*n^3 + 6*n^2 + 7*n + 3 + (n^3 + 3*n^2 - n - 3)*x - (4*n + 4)*x^2 - (n + 1)*x^3)*v[i - 1] + (3*n^2 + 3*n - 3 + (4*n^2 + 4*n - 13)*x + (n^2 + n - 16)*x^2 - 7*x^3 - x^4)*deriv(v[i - 1], x))/(n^3 + 2*n - n*x - n*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-f287bed89e07609f","steps":["basis A013609","D+1"],"edges":["e-7797a63e71923f20","e-143d48ccab264415"],"text":"A125662 \u2192 basis A013609 \u2192 D+1"},"headline":"A125662 \u2192 basis A013609 \u2192 D+1","chain":null}