{"id":"n-85bbd23e228e6595","revision":0,"first_index":0,"fingerprint":"f6-c294b27e66113b73","name":"reversal of basis_map_A013611 of Triangle T(n, k) read by rows: T(n, k) = Sum_{j=0..n} binomial(j, n-k) * |Stirling1(n, n-j)|.","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":true,"rows":[["1"],["4","1"],["32","20","3"],["384","400","136","15"],["6144","9088","4976","1192","105"],["122880","237056","181312","68624","12828","945"],["2949120","7041024","6959104","3641408","1062736","163788","10395"],["82575360","235487232","286388224","192427776","77094912","18401632","2420304","135135"],["2642411520","8774221824","12696731648","10453512192","5353453824","1745275904","353474208","40628880","2027025"],["95126814720","360792981504","606244110336","592170876928","370434044928","153838647552","42394761856","7471701216","763663860","34459425"],["3805072588800","16239128739840","31104831062016","35205473173504","26068608458752","13191792755712","4618724777728","1104368523904","172508877504","15887990340","654729075"],["167423193907200","794428188917760","1709634270265344","2202242271936512","1886333708828672","1127879658885120","480251538089984","145585435384064","30782129612160","4321758002544","362455876440","13749310575"],["8036313307545600","41983286527918080","100334293317844992","145029217269055488","141195590278316032","97523898929545216","48993305982676992","18033886274504704","4826007235217152","915433365201792","116798316127632","8996452065720","316234143225"],["417888291992371200","2384038732140380160","6266965415725891584","10049876630937010176","10967901126198820864","8601132501294252032","4985749384337833984","2162594735841169408","701799533093909504","168252715870921984","28959346568681664","3387773410608240","241355477090700","7905853580625"],["23401744351572787200","144789152883655311360","415319109048440193024","732001157557071642624","885549132102433243136","777796750479846277120","511432543057863507968","255720538584227004416","97690831720970506240","28370739482307187712","6164546736361066752","971617668348466368","104985788803501680","6959325681963900","213458046676875"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","55","935","21505","623645","21827575","894930575","42061737025","2229272062325","131527051677175","8549258359016375","606997343490162625","46738795448742522125","3879320022245629336375"],"nonn":true,"definitions":[{"position":0,"source":"derived:reversal","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","-1"],["4","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":190924,"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-560460f4eb638658"]},{"id":190925,"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-560460f4eb638658"]},{"id":190926,"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-560460f4eb638658"]}],"region_exact":{},"incoming_edges":[{"id":"e-560460f4eb638658","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"reversal\",\"schedule\":{\"first_index\":0,\"first_degree\":0,\"step\":1}}"},"operation_detail":"reversal, shift 0","conclusion_id":"n-85bbd23e228e6595","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":"reversed","lean_declarations":["RealRooted.X_pow_mul_reverse_eq_zero_or_splits","RealRooted.reflect_eq_zero_or_splits"],"certificate_hash":"bf93694a27bf23f5a5ddf3636dbd3d5202643727ad507f84079cebe0343b72dd","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-054881b8a1092a7d","role":"sequence","name":"basis_map_A013611 of Triangle T(n, k) read by rows: T(n, k) = Sum_{j=0..n} binomial(j, n-k) * |Stirling1(n, n-j)|.","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-6d29e559694cc367","kind":"fingerprint_related","node_a_id":"n-054881b8a1092a7d","node_b_id":"n-85bbd23e228e6595","fingerprint":"f6-c294b27e66113b73","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-c294b27e66113b73"},"other_id":"n-054881b8a1092a7d","other":{"id":"n-054881b8a1092a7d","name":"basis_map_A013611 of Triangle T(n, k) read by rows: T(n, k) = Sum_{j=0..n} binomial(j, n-k) * |Stirling1(n, n-j)|.","first_index":0,"fingerprint":"f6-c294b27e66113b73","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","4"],["3","20","32"]],"card_rows":[["1"],["1","4"],["3","20","32"],["15","136","400","384"]],"row_sums_preview":["1","5","55","935","21505","623645","21827575","894930575"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-849de7332df28657","steps":["basis A013611"],"edges":["e-3d3e7193ca830199"],"text":"A088996 \u2192 basis A013611"},"headline":"A088996 \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:           4,            1;\n  2:          32,           20,            3;\n  3:         384,          400,          136,           15;\n  4:        6144,         9088,         4976,         1192,          105;\n  5:      122880,       237056,       181312,        68624,        12828,          945;\n  6:     2949120,      7041024,      6959104,      3641408,      1062736,       163788,       10395;\n  7:    82575360,    235487232,    286388224,    192427776,     77094912,     18401632,     2420304,     135135;\n  8:  2642411520,   8774221824,  12696731648,  10453512192,   5353453824,   1745275904,   353474208,   40628880,   2027025;\n  9: 95126814720, 360792981504, 606244110336, 592170876928, 370434044928, 153838647552, 42394761856, 7471701216, 763663860, 34459425;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (4*n+(2*n-1)*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(4*n + (2*n - 1)*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; for(n = 1, N, my(i = n + 1); v[i] = (4*n + (2*n - 1)*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-849de7332df28657","steps":["basis A013611","reverse"],"edges":["e-3d3e7193ca830199","e-560460f4eb638658"],"text":"A088996 \u2192 basis A013611 \u2192 reverse"},"headline":"A088996 \u2192 basis A013611 \u2192 reverse","chain":null}