{"id":"n-7a54632a32fd58d6","revision":0,"first_index":0,"fingerprint":"f6-2fb4b36f3b7046e1","name":"one_plus_derivative of basis_map_A007318 of Triangle whose (n,k)-th entry is 15^(n-k)*binomial(n,k).","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":true,"rows":[["1"],["17","1"],["288","34","1"],["4864","864","51","1"],["81920","19456","1728","68","1"],["1376256","409600","48640","2880","85","1"],["23068672","8257536","1228800","97280","4320","102","1"],["385875968","161480704","28901376","2867200","170240","6048","119","1"],["6442450944","3087007744","645922816","77070336","5734400","272384","8064","136","1"],["107374182400","57982058496","13891534848","1937768448","173408256","10321920","408576","10368","153","1"],["1786706395136","1073741824000","289910292480","46305116160","4844421120","346816512","17203200","583680","12960","170","1"],["29686813949952","19653770346496","5905580032000","1063004405760","127339069440","10657726464","635830272","27033600","802560","15840","187","1"],["492581209243648","356241767399424","117922622078976","23622320128000","3189013217280","305613766656","21315452928","1089994752","40550400","1070080","19008","204","1"],["8162774324609024","6403555720167424","2315571488096256","510998029008896","76772540416000","8291434364928","662163161088","39585841152","1771241472","58572800","1391104","22464","221","1"],["135107988821114880","114278840544526336","44824890041171968","10806000277782528","1788493101531136","214963113164800","19346680184832","1324326322176","69275222016","2755264512","82001920","1770496","26208","238","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","18","323","5780","103173","1837462","32656711","579301656","10258466825","181369693466","3201872665419","56447829212572","993884992921357","17478667116892830","307041919020084047"],"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":[["15","1"],["1","0"]]},"terms":[{"coeff":{"coeffs":[["256","32","1"],["16","1","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":190659,"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-59c64a4ab7528c0e"]},{"id":190660,"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-59c64a4ab7528c0e"]},{"id":190661,"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-59c64a4ab7528c0e"]},{"id":190662,"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-59c64a4ab7528c0e"]}],"region_exact":{},"incoming_edges":[{"id":"e-59c64a4ab7528c0e","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-7a54632a32fd58d6","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":"ae1e61f39ea27808b5fc2e577bc91b1da33b469d35d8608a6ea41648d1934f2a","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-c20d7dca3713c0eb","role":"sequence","name":"basis_map_A007318 of Triangle whose (n,k)-th entry is 15^(n-k)*binomial(n,k).","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:           17,           1;\n  2:          288,          34,           1;\n  3:         4864,         864,          51,          1;\n  4:        81920,       19456,        1728,         68,         1;\n  5:      1376256,      409600,       48640,       2880,        85,        1;\n  6:     23068672,     8257536,     1228800,      97280,      4320,      102,      1;\n  7:    385875968,   161480704,    28901376,    2867200,    170240,     6048,    119,     1;\n  8:   6442450944,  3087007744,   645922816,   77070336,   5734400,   272384,   8064,   136,   1;\n  9: 107374182400, 57982058496, 13891534848, 1937768448, 173408256, 10321920, 408576, 10368, 153, 1;\n","recurrence_sentence":"The row polynomials satisfy (n+15+x)*P(n,x) = (16*n+256+(n+32)*x+x^2)*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[((16*n + 256 + (n + 32)*x + x^2)*P[n - 1, x])/(n + 15 + x)]];\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] = ((16*n + 256 + (n + 32)*x + x^2)*v[i - 1])/(n + 15 + x)); 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-ff0214c333bfdcca","steps":["x \u21a6 x+1","D+1"],"edges":["e-5b39e32205b8abe3","e-59c64a4ab7528c0e"],"text":"A027467 \u2192 x \u21a6 x+1 \u2192 D+1"},"headline":"A027467 \u2192 x \u21a6 x+1 \u2192 D+1","chain":null}