{"id":"n-9addcdd37c167f65","revision":0,"first_index":0,"fingerprint":"f6-48e472502fc99343","name":"basis_map_A013609 of Triangle, read by rows, T(n, k) = binomial(n, k) for n < 2 and binomial(n, k) + 3^(n-1)*binomial(n-2, k -1) otherwise.","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","2"],["7","14","4"],["26","78","60","8"],["124","496","636","280","16"],["680","3400","6152","4856","1456","32"],["3952","23712","55392","63488","35952","8160","64"],["23456","164192","469248","704320","587680","259296","47552","128"],["140224","1121792","3786304","7012736","7716160","5053184","1826752","281984","256"],["840320","7562880","29411712","64708224","88244352","76487040","41193600","12615552","1684224","512"],["5039872","50398720","221755392","564473856","917285376","987872256","705653760","322609152","85706496","10087936","1024"],["30235136","332586496","1632699392","4716699648","8889203712","11429053440","10159263744","6168225792","2449218048","574541312","60488704","2048"],["181402624","2176831488","11791174656","38094592000","81631365120","121903054848","129522333696","97958449152","51700608000","18140753920","3809639424","362846208","4096"],["1088399360","14149191680","83806758912","299309914112","718344028160","1221185433600","1508524216320","1364856582144","897933256704","419036456960","131697674240","25033635840","2176888832","8192"],["6530363392","91425087488","587732721664","2298688110592","6105890852864","11637111169024","16378159497216","17240172331008","13576640630784","7901752680448","3304371986432","940375932928","163260166144","13060923392","16384"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","25","172","1552","16576","190720","2255872","26939392","322748416","3870883840","46442217472","557273055232","6687142445056","80245172469760"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013609","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["2","2"],["7","14","4"],["26","78","60","8"]],"recurrence":{"denominator":{"coeffs":[["-3"],["1"]]},"terms":[{"coeff":{"coeffs":[["36","72","36"],["-12","-24","-12"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-24","-24"],["8","8"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":199572,"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-d7b86a8b5889d427"]},{"id":199573,"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-d7b86a8b5889d427"]},{"id":199574,"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-d7b86a8b5889d427"]},{"id":199575,"position":3,"claim":"interlacing","from_index":3,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":3,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 3","region_exact":null,"edges":["e-d7b86a8b5889d427"]}],"region_exact":{},"incoming_edges":[{"id":"e-d7b86a8b5889d427","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-9addcdd37c167f65","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":"bf9b29d663ff8ac0e7f454d6e1bcb21ff63ed1b57cf16b3352321502fd7bfea1","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":3,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-fbab854f2622405b","role":"sequence","name":"Triangle, read by rows, T(n, k) = binomial(n, k) for n < 2 and binomial(n, k) + 3^(n-1)*binomial(n-2, k -1) otherwise.","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,       2;\n  2:      7,      14,        4;\n  3:     26,      78,       60,        8;\n  4:    124,     496,      636,      280,       16;\n  5:    680,    3400,     6152,     4856,     1456,       32;\n  6:   3952,   23712,    55392,    63488,    35952,     8160,       64;\n  7:  23456,  164192,   469248,   704320,   587680,   259296,    47552,      128;\n  8: 140224, 1121792,  3786304,  7012736,  7716160,  5053184,  1826752,   281984,     256;\n  9: 840320, 7562880, 29411712, 64708224, 88244352, 76487040, 41193600, 12615552, 1684224, 512;\n","recurrence_sentence":"The row polynomials satisfy (n-3)*P(n,x) = -(12*n-36+(24*n-72)*x+(12*n-36)*x^2)*P(n-2,x) + (8*n-24+(8*n-24)*x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2 + 2*x, P(2,x) = 7 + 14*x + 4*x^2, P(3,x) = 26 + 78*x + 60*x^2 + 8*x^3.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 2*x;\nP[2, x_] := 7 + 14*x + 4*x^2;\nP[3, x_] := 26 + 78*x + 60*x^2 + 8*x^3;\nP[n_, x_] := P[n, x] = Expand[Together[(-(12*n - 36 + (24*n - 72)*x + (12*n - 36)*x^2)*P[n - 2, x] + (8*n - 24 + (8*n - 24)*x)*P[n - 1, x])/(n - 3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 2*x; v[3] = 7 + 14*x + 4*x^2; v[4] = 26 + 78*x + 60*x^2 + 8*x^3; for(n = 4, N, my(i = n + 1); v[i] = (-(12*n - 36 + (24*n - 72)*x + (12*n - 36)*x^2)*v[i - 2] + (8*n - 24 + (8*n - 24)*x)*v[i - 1])/(n - 3)); 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-fbab854f2622405b","steps":["basis A013609"],"edges":["e-d7b86a8b5889d427"],"text":"A146987 \u2192 basis A013609"},"headline":"A146987 \u2192 basis A013609","chain":null}