{"id":"n-5d201923e61d3148","revision":0,"first_index":0,"fingerprint":"f6-a9636442fdb13e67","name":"basis_map_A013611 of (5,1) Pascal triangle.","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"],["6","4"],["12","32","16"],["24","112","160","64"],["48","320","768","768","256"],["96","832","2816","4608","3584","1024"],["192","2048","8960","20480","25600","16384","4096"],["384","4864","26112","76800","133120","135168","73728","16384"],["768","11264","71680","258048","573440","802816","688128","327680","65536"],["1536","25600","188416","802816","2179072","3899392","4587520","3407872","1441792","262144"],["3072","57344","479232","2359296","7569408","16515072","24772608","25165824","16515072","6291456","1048576"],["6144","126976","1187840","6635520","24576000","63307776","115605504","149422080","133693440","78643200","27262976","4194304"],["12288","278528","2883584","18022400","75694080","224919552","484442112","761266176","865075200","692060160","369098752","117440512","16777216"],["24576","606208","6881280","47579136","223477760","752615424","1868562432","3460300800","4775215104","4844421120","3506438144","1711276032","503316480","67108864"],["49152","1310720","16187392","122683392","637272064","2399141888","6747586560","14394851328","23391633408","28789702656","26390560768","17448304640","7851737088","2147483648","268435456"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","10","60","360","2160","12960","77760","466560","2799360","16796160","100776960","604661760","3627970560","21767823360","130606940160"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013611","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["6","4"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["-2","-4"],["2","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":204323,"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-ccaa07d7a5faa89f"]},{"id":204324,"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-ccaa07d7a5faa89f"]},{"id":204325,"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-ccaa07d7a5faa89f"]},{"id":204326,"position":3,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2","region_exact":null,"edges":["e-ccaa07d7a5faa89f"]}],"region_exact":{},"incoming_edges":[{"id":"e-ccaa07d7a5faa89f","operation":{"map":"A013611","operation":"basis_map"},"operation_detail":"A013611","conclusion_id":"n-5d201923e61d3148","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":"efae75f0490e9d31a882271c2ecc4e1d459cb3b1620792e8fb870ce96ea4f41e","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-0c29381e3836fd28","role":"sequence","name":"(5,1) Pascal triangle.","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-72cdeb98e6299ed0","kind":"fingerprint_related","node_a_id":"n-bd7d61845cd3400d","node_b_id":"n-5d201923e61d3148","fingerprint":"f6-a9636442fdb13e67","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-a9636442fdb13e67"},"other_id":"n-bd7d61845cd3400d","other":{"id":"n-bd7d61845cd3400d","name":"basis_map_A013609 of Triangle by columns: (1, 2, 0, 0, 0, ...) in every column.","first_index":0,"fingerprint":"f6-a9636442fdb13e67","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3","2"],["3","8","4"]],"card_rows":[["1"],["3","2"],["3","8","4"],["3","14","20","8"]],"row_sums_preview":["1","5","15","45","135","405","1215","3645"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-e6872af866fa56cc","steps":["basis A013609"],"edges":["e-35dbd4faaf76b3a3"],"text":"A156319 \u2192 basis A013609"},"headline":"A156319 \u2192 basis A013609","chain":null}}],"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,     4;\n  2:   12,    32,     16;\n  3:   24,   112,    160,     64;\n  4:   48,   320,    768,    768,     256;\n  5:   96,   832,   2816,   4608,    3584,    1024;\n  6:  192,  2048,   8960,  20480,   25600,   16384,    4096;\n  7:  384,  4864,  26112,  76800,  133120,  135168,   73728,   16384;\n  8:  768, 11264,  71680, 258048,  573440,  802816,  688128,  327680,   65536;\n  9: 1536, 25600, 188416, 802816, 2179072, 3899392, 4587520, 3407872, 1441792, 262144;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (2*n-2+(4*n-4)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 6 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 6 + 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n - 2 + (4*n - 4)*x)*P[n - 1, x])/(n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 6 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((2*n - 2 + (4*n - 4)*x)*v[i - 1])/(n - 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-0c29381e3836fd28","steps":["basis A013611"],"edges":["e-ccaa07d7a5faa89f"],"text":"A093562 \u2192 basis A013611"},"headline":"A093562 \u2192 basis A013611","chain":null}