{"id":"n-215b7d121fbc1e4b","revision":2,"first_index":0,"fingerprint":"f6-32550b608f1e0aa0","name":"basis_map_A008288 of Triangle of coefficients in expansion of (1+4x)^n.","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"],["5","4"],["25","56","16"],["125","476","368","64"],["625","3376","4704","2048","256"],["3125","21876","45600","35200","10496","1024"],["15625","134376","378096","439808","227072","51200","4096"],["78125","796876","2840592","4522432","3491584","1336320","241664","16384"],["390625","4609376","19953088","40835072","43181568","24453120","7389184","1114112","65536"],["1953125","26171876","133515584","336297728","459240960","354662400","156975104","39059456","5046272","262144"],["9765625","146484376","861328080","2587110400","4374749696","4360851456","2616991744","945356800","199557120","22544384","1048576"],["48828125","810546876","5400390576","18888673088","38336307712","47401687040","36616429568","17828511744","5422252032","992739328","99614720","4194304"],["244140625","4443359376","33095703072","132333985792","314744114944","468448092160","448550584320","280113905664","114194055168","29927407616","4833935360","436207616","16777216"],["1220703125","24169921876","199072265568","896748048512","2452783171328","4292692270080","4950833971200","3825140367360","1992450834432","696443994112","160128040960","23127392256","1895825408","67108864"],["6103515625","130615234376","1178955078064","5912255861248","18317978476800","37001975224320","50254397091840","46736215572480","30045663068160","13369158926336","4081503436800","835102703616","109035126784","8187281408","268435456"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","9","97","1033","11009","117321","1250273","13323977","141991681","1513184649","16125788257","171850175113","1831382268929","19516773915081","207987415033313"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A008288","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["5","4"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-5","12"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["6","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":166220,"position":0,"claim":"real_rooted","from_index":0,"region_key":"[-1564556493/268435456, -46056243/268435456]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1564556493/268435456"},"upper":{"closed":true,"value":"-46056243/268435456"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in [-1564556493/268435456, -46056243/268435456]","region_exact":null,"edges":["e-1d35ea8dc489f8c6"]},{"id":166221,"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-1d35ea8dc489f8c6","e-cfd0ddf0ef7532e7"]},{"id":166222,"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-1d35ea8dc489f8c6","e-cfd0ddf0ef7532e7"]},{"id":168715,"position":3,"claim":"real_rooted","from_index":0,"region_key":"[-1564556493/268435456, -46056243/268435456]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"exact":{"radicand":"2","rational_part":"-3","sqrt_coefficient":"-2"},"value":"-1564556493/268435456"},"upper":{"closed":true,"exact":{"radicand":"2","rational_part":"-3","sqrt_coefficient":"2"},"value":"-46056243/268435456"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in [\u22123\u22122\u221a2, \u22123+2\u221a2] (rational outer bound [-1564556493/268435456, -46056243/268435456])","region_exact":"[\u22123\u22122\u221a2, \u22123+2\u221a2]","edges":["e-cfd0ddf0ef7532e7"]}],"region_exact":{"[-1564556493/268435456, -46056243/268435456]":"[\u22123\u22122\u221a2, \u22123+2\u221a2]"},"incoming_edges":[{"id":"e-1d35ea8dc489f8c6","operation":{"map":"A008288","operation":"basis_map"},"operation_detail":"A008288","conclusion_id":"n-215b7d121fbc1e4b","premise_count":1,"theorem":"affine current multiplier with an affine initial row (module documentation, \"The affine-current theorem\"): with beta = b/d, K = beta^2 + a beta - c > 0 and eta = m/d >= 0, every real polynomial with all roots in E = [-beta - sqrt(K), -beta + sqrt(K)] has a real-rooted image, and no interval of positive length outside E works (Theorem 1); for eta > 0 the images lie in I = [(-a - 2 beta - 2 sqrt(K))/d, (-a - 2 beta + 2 sqrt(K))/d] if and only if eta >= 1/2 and (s - (2 eta - 1) beta)^2 <= (2 eta - 1)^2 K, s = l - eta a (Theorem 2); with a >= 0, K >= beta^2, a + 2 beta >= 0, a^2 + 4c >= 0, l >= 0 and the scalar criterion, PF inputs with roots in [-R, 0], R = beta + sqrt(K), have PF images, sharply (the common-interlacer proof; the Chebyshev case is Iserles and Saff, Math. Proc. Cambridge Philos. Soc. 105 (1989), Theorem 1; the scalar criterion is RealRooted orderTwoConstants_nonneg_iff)","lean_route":null,"lean_declarations":[],"certificate_hash":"3fc30af44c142ded7401fca42b413e8825f668aa38474b955347342b024fc3bd","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1564556493/268435456"},"upper":{"closed":true,"value":"-46056243/268435456"}}}],"premises":[{"position":0,"node_id":"n-2e0409c0014ec240","role":"sequence","name":"Triangle of coefficients in expansion of (1+4x)^n.","uses":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in [-1, 0]"}]},{"id":"e-cfd0ddf0ef7532e7","operation":{"map":"A008288","operation":"basis_map"},"operation_detail":"A008288","conclusion_id":"n-215b7d121fbc1e4b","premise_count":1,"theorem":"affine current multiplier with an affine initial row (module documentation, \"The affine-current theorem\"): with beta = b/d, K = beta^2 + a beta - c > 0 and eta = m/d >= 0, every real polynomial with all roots in E = [-beta - sqrt(K), -beta + sqrt(K)] has a real-rooted image, and no interval of positive length outside E works (Theorem 1); for eta > 0 the images lie in I = [(-a - 2 beta - 2 sqrt(K))/d, (-a - 2 beta + 2 sqrt(K))/d] if and only if eta >= 1/2 and (s - (2 eta - 1) beta)^2 <= (2 eta - 1)^2 K, s = l - eta a (Theorem 2); with a >= 0, K >= beta^2, a + 2 beta >= 0, a^2 + 4c >= 0, l >= 0 and the scalar criterion, PF inputs with roots in [-R, 0], R = beta + sqrt(K), have PF images, sharply (the common-interlacer proof; the Chebyshev case is Iserles and Saff, Math. Proc. Cambridge Philos. Soc. 105 (1989), Theorem 1; the scalar criterion is RealRooted orderTwoConstants_nonneg_iff)","lean_route":null,"lean_declarations":[],"certificate_hash":"6cc3580c7643fe45608384c2fef7b47c154412a39fc68fd0f5b678df821c54ac","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"exact":{"radicand":"2","rational_part":"-3","sqrt_coefficient":"-2"},"value":"-1564556493/268435456"},"upper":{"closed":true,"exact":{"radicand":"2","rational_part":"-3","sqrt_coefficient":"2"},"value":"-46056243/268435456"}}}],"premises":[{"position":0,"node_id":"n-2e0409c0014ec240","role":"sequence","name":"Triangle of coefficients in expansion of (1+4x)^n.","uses":{"claim":"real_rooted","from_index":0,"region":{"lower":{"closed":true,"value":"-1"},"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in [-1, 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:       5,        4;\n  2:      25,       56,        16;\n  3:     125,      476,       368,        64;\n  4:     625,     3376,      4704,      2048,       256;\n  5:    3125,    21876,     45600,     35200,     10496,      1024;\n  6:   15625,   134376,    378096,    439808,    227072,     51200,      4096;\n  7:   78125,   796876,   2840592,   4522432,   3491584,   1336320,    241664,    16384;\n  8:  390625,  4609376,  19953088,  40835072,  43181568,  24453120,   7389184,  1114112,   65536;\n  9: 1953125, 26171876, 133515584, 336297728, 459240960, 354662400, 156975104, 39059456, 5046272, 262144;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(5-12*x)*P(n-2,x) + (6+4*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 5 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 5 + 4*x;\nP[n_, x_] := P[n, x] = Expand[-(5 - 12*x)*P[n - 2, x] + (6 + 4*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 5 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(5 - 12*x)*v[i - 2] + (6 + 4*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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-2e0409c0014ec240","steps":["basis A008288"],"edges":["e-1d35ea8dc489f8c6"],"text":"A013611 \u2192 basis A008288"},"headline":"A013611 \u2192 basis A008288","chain":null}