{"id":"n-3eb516a297f686ad","revision":0,"first_index":0,"fingerprint":"f6-f6df6c2d28e8f6fb","name":"basis_map_A013611 of shift_x_plus_one of Riordan array ((1+x)^3,x).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":3,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["27","108","144","64"],["54","324","720","704","256"],["108","864","2736","4288","3328","1024"],["216","2160","8928","19520","23808","15360","4096"],["432","5184","26496","74752","125696","125952","69632","16384"],["864","12096","73728","255488","550400","754688","643072","311296","65536"],["1728","27648","195840","805888","2122752","3710976","4304896","3194880","1376256","262144"],["3456","62208","502272","2395136","7469056","15912960","23453696","23609344","15532032","6029312","1048576"],["6912","138240","1253376","6799360","24518656","61702144","110559232","141033472","125501440","74186752","26214400","4194304"],["13824","304128","3059712","18612224","76234752","221478912","467927040","724303872","815136768","650379264","349175808","113246208","16777216"],["27648","663552","7335936","49463296","226918400","747896832","1821769728","3320315904","4527489024","4561305600","3299868672","1623195648","486539264","67108864"],["55296","1437696","17326080","128270336","651689984","2403467264","6635126784","13927710720","22336241664","27232567296","24844959744","16445865984","7465861120","2080374784","268435456"],["110592","3096576","40402944","325844992","1816461312","7413694464","22884122624","54395928576","100383326208","143810101248","158620188672","132271570944","80715186176","34024194048","8858370048","1073741824"],["221184","6635520","93192192","813301760","4936302592","22093234176","75423023104","200328347648","418350366720","689153507328","892480782336","899023896576","690516656128","390909132800","153813516288","37580963840","4294967296"],["442368","14155776","212926464","1999372288","13125812224","63931678720","239218982912","702348787712","1638014124032","3051708481536","4541575593984","5367970922496","4977128898560","3543884890112","1871263563776","690415992832","158913789952","17179869184"]],"degrees":[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17],"row_sums":["343","2058","12348","74088","444528","2667168","16003008","96018048","576108288","3456649728","20739898368","124439390208","746636341248","4479818047488","26878908284928"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013611","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["27","108","144","64"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":204612,"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-2c0e83fb0e1d16d7"]},{"id":204613,"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-2c0e83fb0e1d16d7"]},{"id":204614,"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-2c0e83fb0e1d16d7"]}],"region_exact":{},"incoming_edges":[{"id":"e-2c0e83fb0e1d16d7","operation":{"map":"A013611","operation":"basis_map"},"operation_detail":"A013611","conclusion_id":"n-3eb516a297f686ad","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":"ff8b1c38d8dbe7ef8f322d4df159b282798fd8a31ba9d22879c2f12956709faa","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-9aa9e3dd2403bca2","role":"sequence","name":"shift_x_plus_one of Riordan array ((1+x)^3,x).","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:    27,    108,     144,       64;\n  1:    54,    324,     720,      704,      256;\n  2:   108,    864,    2736,     4288,     3328,      1024;\n  3:   216,   2160,    8928,    19520,    23808,     15360,      4096;\n  4:   432,   5184,   26496,    74752,   125696,    125952,     69632,     16384;\n  5:   864,  12096,   73728,   255488,   550400,    754688,    643072,    311296,     65536;\n  6:  1728,  27648,  195840,   805888,  2122752,   3710976,   4304896,   3194880,   1376256,    262144;\n  7:  3456,  62208,  502272,  2395136,  7469056,  15912960,  23453696,  23609344,  15532032,   6029312,   1048576;\n  8:  6912, 138240, 1253376,  6799360, 24518656,  61702144, 110559232, 141033472, 125501440,  74186752,  26214400,   4194304;\n  9: 13824, 304128, 3059712, 18612224, 76234752, 221478912, 467927040, 724303872, 815136768, 650379264, 349175808, 113246208, 16777216;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2+4*x)*P(n-1,x) for n >= 1, with P(0,x) = 27 + 108*x + 144*x^2 + 64*x^3.","mathematica":"P[0, x_] := 27 + 108*x + 144*x^2 + 64*x^3;\nP[n_, x_] := P[n, x] = Expand[(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] = 27 + 108*x + 144*x^2 + 64*x^3; for(n = 1, N, my(i = n + 1); v[i] = (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-f48bed27dbc29c5f","steps":["P\u2099 + P\u2099\u208a\u2081","basis A013611"],"edges":["e-0331d381cbbd8a41","e-2c0e83fb0e1d16d7"],"text":"A391533 \u2192 P\u2099 + P\u2099\u208a\u2081 \u2192 basis A013611"},"headline":"A391533 \u2192 P\u2099 + P\u2099\u208a\u2081 \u2192 basis A013611","chain":null}
