{"id":"n-f6b16b0bcda6a25a","revision":0,"first_index":1,"fingerprint":"f6-b860ced141a23493","name":"basis_map_A013609 of basis_map_A013611 of Triangle read by rows: T(n, k) = binomial(2n, 2k).","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"],["6","8"],["56","128","64"],["576","1920","1920","512"],["6016","26624","40960","24576","4096"],["62976","348160","727040","696320","286720","32768"],["659456","4374528","11550720","15269888","10321920","3145728","262144"],["6905856","53444608","170655744","288096256","272498688","139460608","33030144","2097152"],["72318976","639631360","2395996160","4924112896","5997854720","4345298944","1761607680","335544320","16777216"],["757334016","7535591424","32393134080","78473330688","117039955968","109999816704","63770198016","21139292160","3321888768","134217728"],["7930904576","87681925120","425386311680","1186904145920","2095600107520","2424106385408","1835091886080","877783941120","243605176320","32212254720","1073741824"],["83053510656","1010038734848","5458509168640","17238019014656","35169574584320","48346030931968","45255936180224","28435367854080","11480447582208","2716566814720","307090161664","8589934592"],["869747654656","11538832293888","68737327693824","242365541384192","561127008239616","893999590146048","997985043349504","778686160699392","415209520889856","144001663500288","29480655519744","2886218022912","68719476736"],["9108115685376","130905905561600","852175198617600","3318388369653760","8596081763942400","15578404728340480","20210720230604800","18866502841139200","12541562202357760","5773295039283200","1744272118251520","312673619148800","26800595927040","549755813888"],["95381425750016","1476315551105024","10426650512588800","44439940712366080","127358579328942080","258750648344903680","382687255430430720","416223562668441600","332012482493153280","191231185471406080","77042779758264320","20511389416161280","3251805639147520","246290604621824","4398046511104"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","14","248","4928","102272","2153984","45584384","966189056","20489142272","434564759552","9217376780288","195509224472576","4146956028870656","87961440484327424","1865758665403793408"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013609","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["6","8"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-16","-64","-64"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["12","16"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":200429,"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-3c3a8a07e399da70"]},{"id":200430,"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-3c3a8a07e399da70"]},{"id":200431,"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-3c3a8a07e399da70"]},{"id":200432,"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-3c3a8a07e399da70"]}],"region_exact":{},"incoming_edges":[{"id":"e-3c3a8a07e399da70","operation":{"map":"A013609","operation":"basis_map"},"operation_detail":"A013609","conclusion_id":"n-f6b16b0bcda6a25a","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":"a251ab3242fd37f24ba01268afc78dd34c2d08ca1af1586eefe819a5680ffb73","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-7ff58d52d83b38f4","role":"sequence","name":"basis_map_A013611 of Triangle read by rows: T(n, k) = binomial(2n, 2k).","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":[{"id":"l-39617034eeecb260","kind":"fingerprint_related","node_a_id":"n-12a33c74cd3c78c7","node_b_id":"n-f6b16b0bcda6a25a","fingerprint":"f6-b860ced141a23493","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-b860ced141a23493"},"other_id":"n-12a33c74cd3c78c7","other":{"id":"n-12a33c74cd3c78c7","name":"basis_map_A013609 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).","first_index":0,"fingerprint":"f6-b860ced141a23493","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","4"],["7","16","8"]],"card_rows":[["1"],["3","4"],["7","16","8"],["18","60","60","16"]],"row_sums_preview":["1","7","31","154","799","4207","22258","117943"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2496e2f555756362","steps":["basis A013609"],"edges":["e-7e038de2d24440b9"],"text":"A084534 \u2192 basis A013609"},"headline":"A084534 \u2192 basis A013609","chain":null}}],"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:         6,          8;\n   3:        56,        128,          64;\n   4:       576,       1920,        1920,         512;\n   5:      6016,      26624,       40960,       24576,         4096;\n   6:     62976,     348160,      727040,      696320,       286720,        32768;\n   7:    659456,    4374528,    11550720,    15269888,     10321920,      3145728,      262144;\n   8:   6905856,   53444608,   170655744,   288096256,    272498688,    139460608,    33030144,     2097152;\n   9:  72318976,  639631360,  2395996160,  4924112896,   5997854720,   4345298944,  1761607680,   335544320,   16777216;\n  10: 757334016, 7535591424, 32393134080, 78473330688, 117039955968, 109999816704, 63770198016, 21139292160, 3321888768, 134217728;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(16+64*x+64*x^2)*P(n-2,x) + (12+16*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 6 + 8*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 6 + 8*x;\nP[n_, x_] := P[n, x] = Expand[-(16 + 64*x + 64*x^2)*P[n - 2, x] + (12 + 16*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 6 + 8*x; for(n = 3, N, my(i = n); v[i] = -(16 + 64*x + 64*x^2)*v[i - 2] + (12 + 16*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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-f7935c36a3c0ec8c","steps":["basis A013611","basis A013609"],"edges":["e-11143fa0a471b62a","e-3c3a8a07e399da70"],"text":"A086645 \u2192 basis A013611 \u2192 basis A013609"},"headline":"A086645 \u2192 basis A013611 \u2192 basis A013609","chain":null}