{"id":"n-75e61a8444ba7f16","revision":1,"first_index":0,"fingerprint":"f6-621e311c1d8dbd55","name":"basis_map_A013611 of Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + (n^2 +n -1)*T(n-2, k-1), read by rows.","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","4"],["9","36","16"],["40","240","352","64"],["251","2008","4704","2752","256"],["1662","16620","55488","67008","18944","1024"],["13615","163380","699632","1240256","767744","119808","4096"],["118640","1660960","8761024","21171840","22280192","7534592","712704","16384"],["1203945","19263120","120239680","364141440","535990272","330735616","65880064","4046848","65536"],["12966850","233403300","1688564736","6212459904","12048658432","11407759360","4200448000","526794752","22151168","262144"],["157163705","3143274100","25815588112","111295096192","266135763968","340752093184","205413588992","47020195840","3923050496","117702656","1048576"],["2012984760","44285664720","407709582240","2024492713920","5811155156992","9553960736768","8301760143360","3185739612160","474869137408","27569160192","610271232","4194304"],["28386343795","681272251080","6942807922720","38936228296000","129974284223744","260173290205184","297924767301632","174223075328000","43453290971136","4405150089216","184635359232","3099590656","16777216"],["421122929150","10949196157900","122468960492480","767258607044160","2933245289925120","6976786944322560","10056228854792192","8127238433456128","3176225676328960","531418737541120","38060402147328","1187336159232","15468593152","67108864"],["6774991711455","189699767920740","2309325163367920","15966252193316160","68650837266245376","188721274312279040","327790752464449536","341957120493797376","193593633873657856","51015367800913920","5923090019123200","309619255672832","7375046049792","76034342912","268435456"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","6","61","696","9971","160746","3008531","62256336","1441566521","36353468646","1003774565821","29834169358056","956930304659611","32741505096998346","1196441198951283571"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013611","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["2","4"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-1","-4"],["1","4"],["1","4"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["2","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":168093,"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-995122d0643acde9"]},{"id":168094,"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-995122d0643acde9"]},{"id":168095,"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-995122d0643acde9"]},{"id":168096,"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-995122d0643acde9"]}],"region_exact":{},"incoming_edges":[{"id":"e-995122d0643acde9","operation":{"map":"A013611","operation":"basis_map"},"operation_detail":"A013611","conclusion_id":"n-75e61a8444ba7f16","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":"22eb94c56b00362bdc487954ba2c8ae8f6bb0236d7e29f8964b7343a2d94b6ac","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-74f9a0f107d622a7","role":"sequence","name":"Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + (n^2 +n -1)*T(n-2, k-1), read by rows.","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,         4;\n  2:        9,        36,         16;\n  3:       40,       240,        352,         64;\n  4:      251,      2008,       4704,       2752,         256;\n  5:     1662,     16620,      55488,      67008,       18944,        1024;\n  6:    13615,    163380,     699632,    1240256,      767744,      119808,       4096;\n  7:   118640,   1660960,    8761024,   21171840,    22280192,     7534592,     712704,     16384;\n  8:  1203945,  19263120,  120239680,  364141440,   535990272,   330735616,   65880064,   4046848,    65536;\n  9: 12966850, 233403300, 1688564736, 6212459904, 12048658432, 11407759360, 4200448000, 526794752, 22151168, 262144;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (n^2+n-1+(4*n^2+4*n-4)*x)*P(n-2,x) + (2+4*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 4*x;\nP[n_, x_] := P[n, x] = Expand[(n^2 + n - 1 + (4*n^2 + 4*n - 4)*x)*P[n - 2, x] + (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] = 1; v[2] = 2 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (n^2 + n - 1 + (4*n^2 + 4*n - 4)*x)*v[i - 2] + (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-74f9a0f107d622a7","steps":["basis A013611"],"edges":["e-995122d0643acde9"],"text":"A154233 \u2192 basis A013611"},"headline":"A154233 \u2192 basis A013611","chain":null}