{"id":"n-300f32295f9afdcb","revision":0,"first_index":1,"fingerprint":"f6-d0568900822197f8","name":"basis_map_A008277 of basis_map_A013611 of Triangle read by rows: T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["8","4"],["55","76","16"],["377","1052","544","64"],["2584","13004","12352","3392","256"],["17711","152328","236752","113792","19456","1024"],["121393","1733448","4148240","3105664","902400","105472","4096"],["832040","19401172","68920640","75486784","33158400","6456320","548864","16384"],["5702887","215007604","1107266048","1706825792","1063687936","306848768","42835968","2768896","65536"],["39088169","2368494020","17400100496","36783267456","31258967296","12533212160","2554761216","268238848","13631488","262144"],["267914296","25994779292","269368217472","766775657856","865025629696","463125748736","129287610368","19604635648","1604517888","65798144","1048576"],["1836311903","284645735376","4127102278752","15609427101696","22931433147136","15951964522496","5857864155136","1203044450304","140981108736","9247129600","312475648","4194304"],["12586269025","3112453557456","62775748905312","312320748389376","588995645161216","521965648947200","245272359288832","65518407745536","10308836524032","961485209600","51673825280","1463812096","16777216"],["86267571272","34002688472740","949958945530752","6169587787917696","14773251399118336","16433103137696768","9685447869587456","3273030663798784","663977274310656","82569962979328","6274298150912","281345523712","6777995264","67108864"],["591286729879","371262189927772","14322619187946928","120709683004169856","363874137686231296","502264824665338880","365736024148709376","153232346303561728","38976927516786688","6205456020144128","625120191184896","39444350500864","1498155057152","31071404032","268435456"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","12","147","2037","31588","541063","10120713","204820604","4451009435","103220023293","2541121557972","66117862611087","1811297074412177","52071578485762540","1566359966046128931"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A008277","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["8","4"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["7","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":197128,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-ea72727a19212e27"]},{"id":197129,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-ea72727a19212e27"]},{"id":197130,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-ea72727a19212e27"]}],"region_exact":{},"incoming_edges":[{"id":"e-ea72727a19212e27","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-300f32295f9afdcb","premise_count":1,"theorem":"ray basis maps: for L f = (a + b t) f + kappa (t - h) f', b > 0, kappa >= 0, and g real-rooted with every root <= h, of multiplicity m at h, T(u^k) = L^k g sends every real polynomial with all roots in S to one with all roots in (-inf, h] if and only if S lies in (-inf, a + b h + kappa m] (RealRooted rayBasisMap_maps_realRootedIn_iff; the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"2253c9f7b3fe2748159776721193ca395acf3bfa6949ca3348704570350bd5d1","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-0e9006d690eb01e9","role":"sequence","name":"basis_map_A013611 of Triangle read by rows: T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:        1;\n   2:        8,          4;\n   3:       55,         76,          16;\n   4:      377,       1052,         544,          64;\n   5:     2584,      13004,       12352,        3392,         256;\n   6:    17711,     152328,      236752,      113792,       19456,        1024;\n   7:   121393,    1733448,     4148240,     3105664,      902400,      105472,       4096;\n   8:   832040,   19401172,    68920640,    75486784,    33158400,     6456320,     548864,     16384;\n   9:  5702887,  215007604,  1107266048,  1706825792,  1063687936,   306848768,   42835968,   2768896,    65536;\n  10: 39088169, 2368494020, 17400100496, 36783267456, 31258967296, 12533212160, 2554761216, 268238848, 13631488, 262144;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -P(n-2,x) + (7+4*x)*P(n-1,x) + 4*x*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 8 + 4*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 8 + 4*x;\nP[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (7 + 4*x)*P[n - 1, x] + 4*x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 8 + 4*x; for(n = 3, N, my(i = n); v[i] = -v[i - 2] + (7 + 4*x)*v[i - 1] + 4*x*deriv(v[i - 1], x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-318700a428d7b97b","steps":["basis A013611","basis A008277"],"edges":["e-50e29512dcce7c20","e-ea72727a19212e27"],"text":"A111125 \u2192 basis A013611 \u2192 basis A008277"},"headline":"A111125 \u2192 basis A013611 \u2192 basis A008277","chain":null}