{"id":"n-920469c0112e70be","revision":1,"first_index":0,"fingerprint":"f6-73de6fe2aa6dc63d","name":"basis_map_A013611 of Pascal-(1,7,1) array.","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"],["2","4"],["11","44","16"],["36","216","320","64"],["149","1192","2848","1856","256"],["550","5500","18752","24512","9728","1024"],["2143","25716","112816","216768","171264","48128","4096"],["8136","113904","613760","1581440","1964032","1060864","229376","16384"],["31273","500368","3192896","10294144","17522176","15110144","6078464","1064960","65536"],["119498","2150964","15872896","61615232","134249472","162727936","103907328","32980992","4849664","262144"],["457907","9158140","76710224","348182144","925851136","1458845696","1324359680","659243008","171900928","21757952","1048576"],["1752300","38550600","361390272","1878952896","5909403648","11519176704","13767835648","9756196864","3938189312","868745216","96468992","4194304"],["6709949","161038776","1670382432","9788628160","35564676864","82811719680","123730575360","116280508416","66563276800","22455779328","4284481536","423624704","16777216"],["25685998","667835948","7594068672","49530383936","204260372992","553979685888","997619826688","1181276094464","898989424640","427515183104","123392229376","20715667456","1845493760","67108864"],["98341639","2753565892","34061243888","244728147776","1129676608256","3500493853696","7396000575488","10631451164672","10244880400384","6471950270464","2615598383104","657931108352","98532589568","7985954816","268435456"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","6","71","636","6301","60066","580931","5587896","53859961","518736126","4997515391","48140856756","463758179221","4467479061786","43036410643451"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013611","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["2","4"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["7","28"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["2","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":167906,"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-85efc751f4220bb8"]},{"id":167907,"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-85efc751f4220bb8"]},{"id":167908,"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-85efc751f4220bb8"]},{"id":167909,"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-85efc751f4220bb8"]}],"region_exact":{},"incoming_edges":[{"id":"e-85efc751f4220bb8","operation":{"map":"A013611","operation":"basis_map"},"operation_detail":"A013611","conclusion_id":"n-920469c0112e70be","premise_count":1,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"f20548f25bbaece60cfad25969671cdd9c0a3f9a1949bad8479899b8d2280788","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-1e2f1b7f566b371c","role":"sequence","name":"Pascal-(1,7,1) array.","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:      2,       4;\n  2:     11,      44,       16;\n  3:     36,     216,      320,       64;\n  4:    149,    1192,     2848,     1856,       256;\n  5:    550,    5500,    18752,    24512,      9728,      1024;\n  6:   2143,   25716,   112816,   216768,    171264,     48128,      4096;\n  7:   8136,  113904,   613760,  1581440,   1964032,   1060864,    229376,    16384;\n  8:  31273,  500368,  3192896, 10294144,  17522176,  15110144,   6078464,  1064960,   65536;\n  9: 119498, 2150964, 15872896, 61615232, 134249472, 162727936, 103907328, 32980992, 4849664, 262144;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (7+28*x)*P(n-2,x) + (2+4*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 4*x;\nP[n_, x_] := P[n, x] = Expand[(7 + 28*x)*P[n - 2, x] + (2 + 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] = 2 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (7 + 28*x)*v[i - 2] + (2 + 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-1e2f1b7f566b371c","steps":["basis A013611"],"edges":["e-85efc751f4220bb8"],"text":"A081582 \u2192 basis A013611"},"headline":"A081582 \u2192 basis A013611","chain":null}
