{"id":"n-3a1bf08f9f4a4163","revision":1,"first_index":0,"fingerprint":"f6-ea8a87f06e938817","name":"basis_map_A013610 of Pascal-(1,2,1) array read by antidiagonals.","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","3"],["6","18","9"],["16","72","90","27"],["44","264","522","378","81"],["120","900","2448","2916","1458","243"],["328","2952","10224","17064","14094","5346","729"],["896","9408","39600","85320","99792","62208","18954","2187"],["2448","29376","145584","384912","586116","519048","258066","65610","6561"],["6688","90288","514944","1614816","3038472","3519612","2484432","1023516","223074","19683"],["18272","274080","1768176","6417792","14403096","20709432","19158120","11179944","3923478","747954","59049"],["49920","823680","5930208","24459408","63825408","109898208","126531072","96787872","47974032","14644152","2480058","177147"],["136384","2454912","19512288","90154080","268341984","539110080","745329600","710477568","461238300","198247176","53498394","8148762","531441"],["372608","7265856","63191808","323345088","1081553472","2485994976","4020440832","4609706112","3730584600","2097341748","794563344","192027348","26572050","1594323"],["1017984","21377664","201935232","1133647488","4210750656","10904922432","20224186272","27173667456","26475629544","18550361448","9177631776","3105032616","679181598","86093442","4782969"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","33","205","1289","8085","50737","318365","1997721","12535525","78659393","493581165","3097180969","19434554165","121950218577"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013610","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["2","3"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","6"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["2","3"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":167087,"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-eece50974f996e38"]},{"id":167088,"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-eece50974f996e38"]},{"id":167089,"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-eece50974f996e38"]},{"id":167090,"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-eece50974f996e38"]}],"region_exact":{},"incoming_edges":[{"id":"e-eece50974f996e38","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-3a1bf08f9f4a4163","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":"cedee635cf4f23d5005202a3176ed74e4e924972908f18f3879c573b5ead8ce1","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-f25cd955a876cde3","role":"sequence","name":"Pascal-(1,2,1) array read by antidiagonals.","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":[{"id":"l-dd10bf317e5d12be","kind":"oeis_match","node_a_id":"n-3a1bf08f9f4a4163","node_b_id":null,"fingerprint":null,"oeis_id":"A015544","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A015544","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A015544","relation":"match_row_sums"}],"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,     3;\n  2:    6,    18,      9;\n  3:   16,    72,     90,      27;\n  4:   44,   264,    522,     378,      81;\n  5:  120,   900,   2448,    2916,    1458,     243;\n  6:  328,  2952,  10224,   17064,   14094,    5346,     729;\n  7:  896,  9408,  39600,   85320,   99792,   62208,   18954,    2187;\n  8: 2448, 29376, 145584,  384912,  586116,  519048,  258066,   65610,   6561;\n  9: 6688, 90288, 514944, 1614816, 3038472, 3519612, 2484432, 1023516, 223074, 19683;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2+6*x)*P(n-2,x) + (2+3*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 3*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 3*x;\nP[n_, x_] := P[n, x] = Expand[(2 + 6*x)*P[n - 2, x] + (2 + 3*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 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = (2 + 6*x)*v[i - 2] + (2 + 3*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-f25cd955a876cde3","steps":["basis A013610"],"edges":["e-eece50974f996e38"],"text":"A081577 \u2192 basis A013610"},"headline":"A081577 \u2192 basis A013610","chain":null}
