{"id":"n-8018092894e25502","revision":0,"first_index":0,"fingerprint":"f6-261d320b22506b21","name":"basis_map_A013611 of Triangle read by rows: T(n,0)=1, T(n,n)=2, T(n,m) = T(n-1,m-1) + Sum_{k=0..m} T(n-1-k,m-k).","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"],["3","8"],["7","32","32"],["18","120","240","128"],["47","416","1280","1536","512"],["123","1360","5680","10880","8960","2048"],["322","4272","22560","59648","80640","49152","8192"],["843","13048","83328","281344","532224","544768","258048","32768"],["2207","39040","292480","1202176","2928640","4243456","3440640","1310720","131072"],["5778","114984","988560","4789632","14287104","26855424","31137792","20643840","6488064","524288"],["15127","334480","3245440","18110720","63952640","147955712","224010240","214302720","118947840","31457280","2097152"],["39603","963248","10411280","65757824","268322560","737701888","1381101568","1735557120","1401421824","663224320","149946368","8388608"],["103682","2751072","32776512","231137792","1070264832","3410337792","7614021632","11881807872","12671188992","8789164032","3598712832","704643072","33554432"],["271443","7802600","101587200","791165440","4098931200","14856724480","38548889600","71969996800","95684526080","88093491200","53230960640","19084083200","3271557120","134217728"],["710647","21998816","310738400","2648826880","15182325760","61690961920","182479503360","396941721600","633263554560","729489080320","587789762560","312978964480","99237232640","15032385536","536870912"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","11","71","506","3791","29051","224786","1746371","13590431","105835466","824429351","6422836211","50040464546","389874204731","3037604638391"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013611","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["3","8"],["7","32","32"]],"recurrence":{"denominator":{"coeffs":[["-2"],["1"]]},"terms":[{"coeff":{"coeffs":[["2","16","32"],["-1","-8","-16"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-6","-16"],["3","8"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":204540,"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-558ac76b5d434951"]},{"id":204541,"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-558ac76b5d434951"]},{"id":204542,"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-558ac76b5d434951"]},{"id":204543,"position":3,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2","region_exact":null,"edges":["e-558ac76b5d434951"]}],"region_exact":{},"incoming_edges":[{"id":"e-558ac76b5d434951","operation":{"map":"A013611","operation":"basis_map"},"operation_detail":"A013611","conclusion_id":"n-8018092894e25502","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":"2275b4279d82b7393a3cbf93a7464a79bb8f7e339cf5d6338882bae28da081e2","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-2496e2f555756362","role":"sequence","name":"Triangle read by rows: T(n,0)=1, T(n,n)=2, T(n,m) = T(n-1,m-1) + Sum_{k=0..m} T(n-1-k,m-k).","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-78e8b87dec95bf98","kind":"fingerprint_related","node_a_id":"n-8018092894e25502","node_b_id":"n-3076284606182517","fingerprint":"f6-261d320b22506b21","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-261d320b22506b21"},"other_id":"n-3076284606182517","other":{"id":"n-3076284606182517","name":"basis_map_A013611 of basis_map_A013611 of Triangle read by rows: T(n, k) = binomial(2n, 2k).","first_index":1,"fingerprint":"f6-261d320b22506b21","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["6","16"],["56","256","256"]],"card_rows":[["1"],["6","16"],["56","256","256"],["576","3840","7680","4096"]],"row_sums_preview":["1","22","568","16192","485248","14874112","460361728","14306271232"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-f7935c36a3c0ec8c","steps":["basis A013611","basis A013611"],"edges":["e-11143fa0a471b62a","e-e91d93c2a0ac15aa"],"text":"A086645 \u2192 basis A013611 \u2192 basis A013611"},"headline":"A086645 \u2192 basis A013611 \u2192 basis A013611","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:    3,      8;\n  2:    7,     32,     32;\n  3:   18,    120,    240,     128;\n  4:   47,    416,   1280,    1536,      512;\n  5:  123,   1360,   5680,   10880,     8960,     2048;\n  6:  322,   4272,  22560,   59648,    80640,    49152,     8192;\n  7:  843,  13048,  83328,  281344,   532224,   544768,   258048,    32768;\n  8: 2207,  39040, 292480, 1202176,  2928640,  4243456,  3440640,  1310720,  131072;\n  9: 5778, 114984, 988560, 4789632, 14287104, 26855424, 31137792, 20643840, 6488064, 524288;\n","recurrence_sentence":"The row polynomials satisfy (n-2)*P(n,x) = -(n-2+(8*n-16)*x+(16*n-32)*x^2)*P(n-2,x) + (3*n-6+(8*n-16)*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 3 + 8*x, P(2,x) = 7 + 32*x + 32*x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 3 + 8*x;\nP[2, x_] := 7 + 32*x + 32*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n - 2 + (8*n - 16)*x + (16*n - 32)*x^2)*P[n - 2, x] + (3*n - 6 + (8*n - 16)*x)*P[n - 1, x])/(n - 2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 8*x; v[3] = 7 + 32*x + 32*x^2; for(n = 3, N, my(i = n + 1); v[i] = (-(n - 2 + (8*n - 16)*x + (16*n - 32)*x^2)*v[i - 2] + (3*n - 6 + (8*n - 16)*x)*v[i - 1])/(n - 2)); 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-2496e2f555756362","steps":["basis A013611"],"edges":["e-558ac76b5d434951"],"text":"A084534 \u2192 basis A013611"},"headline":"A084534 \u2192 basis A013611","chain":null}