{"id":"n-38b2cd85e152cf9d","revision":0,"first_index":0,"fingerprint":"f6-7e8ebe3586b974d9","name":"basis_map_A013611 of basis_map_A013609 of Triangle of coefficients of Chebyshev's S(n,x+4) polynomials (exponents of x in increasing order).","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"],["7","8"],["48","112","64"],["329","1160","1344","512"],["2255","10640","18624","14336","4096"],["15456","91360","214144","248832","143360","32768"],["105937","752528","2211264","3440640","2990080","1376256","262144"],["726103","6023832","21284928","41525760","48312320","33521664","12845056","2097152"],["4976784","47223120","194973888","457519104","667402240","619773952","357826560","117440512","16777216"],["34111385","364352280","1721317248","4720899072","8283656192","9644113920","7450132480","3682598912","1056964608","134217728"],["233802911","2776133920","14769065088","46359312384","95085383680","133158273024","128946012160","85261811712","36842766336","9395240960","1073741824"],["1602508992","20939008448","123871209728","437946808320","1028188528640","1683146866688","1960438136832","1624718180352","938936893440","360374599680","82678120448","8589934592"],["10983760033","156616997152","1019841470592","4010238023680","10605808783360","19874378022912","27059295879168","26971270545408","19533460930560","10024722104320","3460669898752","721554505728","68719476736"],["75283811239","1163250051880","8267955061632","35792451122176","105294377144320","222283969560576","346449657200640","403648542670848","351565453983744","226080367575040","104339788005376","32727650795520","6253472382976","549755813888"],["516002918640","8588403855920","66161844375872","312680560324608","1012794440204288","2378468426055680","4176360061009920","5570165785755648","5670613058322432","4385061482790912","2535560786739200","1063090305105920","305526793568256","53876069761024","4398046511104"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","15","224","3345","49951","745920","11138849","166336815","2483913376","37092363825","553901543999","8271430796160","123517560398401","1844491975179855","27543862067299424"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013611","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["7","8"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["7","8"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":205715,"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-e2a754028f824170"]},{"id":205716,"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-e2a754028f824170"]},{"id":205717,"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-e2a754028f824170"]},{"id":205718,"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-e2a754028f824170"]}],"region_exact":{},"incoming_edges":[{"id":"e-e2a754028f824170","operation":{"map":"A013611","operation":"basis_map"},"operation_detail":"A013611","conclusion_id":"n-38b2cd85e152cf9d","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":"485458059c78fdd32f2c7e9731b93a9e7c08ddba47aad83d2b6e57e7edb60500","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-4f1c1e4637c4f22d","role":"sequence","name":"basis_map_A013609 of Triangle of coefficients of Chebyshev's S(n,x+4) polynomials (exponents of x in increasing order).","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-d8903784d98edc65","kind":"fingerprint_related","node_a_id":"n-36eb2fa2de362ed0","node_b_id":"n-38b2cd85e152cf9d","fingerprint":"f6-7e8ebe3586b974d9","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-7e8ebe3586b974d9"},"other_id":"n-36eb2fa2de362ed0","other":{"id":"n-36eb2fa2de362ed0","name":"basis_map_A013609 of basis_map_A013611 of Triangle read by rows: T(n, k) = binomial(n+k-1, 2*k-1).","first_index":1,"fingerprint":"f6-7e8ebe3586b974d9","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["7","8"],["48","112","64"]],"card_rows":[["1"],["7","8"],["48","112","64"],["329","1160","1344","512"]],"row_sums_preview":["1","15","224","3345","49951","745920","11138849","166336815"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-976cdb6e4a693ec8","steps":["basis A013611","basis A013609"],"edges":["e-60bc1ac61eca3352","e-e31051ac63013a05"],"text":"A078812 \u2192 basis A013611 \u2192 basis A013609"},"headline":"A078812 \u2192 basis A013611 \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:        7,         8;\n  2:       48,       112,         64;\n  3:      329,      1160,       1344,        512;\n  4:     2255,     10640,      18624,      14336,       4096;\n  5:    15456,     91360,     214144,     248832,     143360,      32768;\n  6:   105937,    752528,    2211264,    3440640,    2990080,    1376256,     262144;\n  7:   726103,   6023832,   21284928,   41525760,   48312320,   33521664,   12845056,    2097152;\n  8:  4976784,  47223120,  194973888,  457519104,  667402240,  619773952,  357826560,  117440512,   16777216;\n  9: 34111385, 364352280, 1721317248, 4720899072, 8283656192, 9644113920, 7450132480, 3682598912, 1056964608, 134217728;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -P(n-2,x) + (7+8*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 7 + 8*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 7 + 8*x;\nP[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (7 + 8*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] = 7 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = -v[i - 2] + (7 + 8*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-30cb3cad216ed189","steps":["basis A013609","basis A013611"],"edges":["e-7ff4c96228a6f1e1","e-e2a754028f824170"],"text":"A207823 \u2192 basis A013609 \u2192 basis A013611"},"headline":"A207823 \u2192 basis A013609 \u2192 basis A013611","chain":null}