{"id":"n-946a37787f674849","revision":1,"first_index":0,"fingerprint":"f6-4dd3c5f9ea752f04","name":"basis_map_A008277 of Triangle read by rows: T(n,k) = |s(n,n+1-k)|, where s(n,k) are the signed Stirling numbers of the first kind A008276 (1 <= k <= n; in other words, the unsigned Stirling numbers of the first kind in reverse order).","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","1"],["6","9","2"],["24","81","47","6"],["120","825","935","290","24"],["720","9675","19085","10765","2074","120"],["5040","130095","420665","383635","128884","16884","720"],["40320","1986705","10165295","14070070","7325269","1628200","154188","5040"],["362880","34096545","270026015","545634670","412896589","138383952","21814316","1561104","40320"],["3628800","651100275","7867597325","22618716925","23903955124","11327186661","2641571792","310289436","17356176","362880"],["39916800","13709393775","250406519825","1006043367175","1445288880574","929997937561","300823463842","51549162236","4682306376","210212640","3628800"],["479001600","315755141625","8664625005575","48026423240950","92002654351069","78208039502901","33694207492847","7896933541266","1035271422196","74838973176","2755052640","39916800"],["6227020800","7899626559825","324380186905775","2457270238873450","6188478994306609","6813218735924601","3802504110926807","1170333043416966","207607787514556","21478772820648","1264489679232","38848567680","479001600"],["87178291200","213370868385675","13077902621508725","134452265090067925","440384126729591044","618694507064029396","438402221772286057","172317523000829921","39712048491138166","5512620477194116","461310560477832","22537590955776","586461651840","6227020800"],["1307674368000","6188975679261375","565337004882274625","7846869746834993875","33149604709061915074","58754410874656773276","52063543054352731187","25609511203905785871","7455875480386165976","1333248166160304680","148679823554781584","10267200718215792","422848755667584","9437186223360","87178291200"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","3","17","158","2194","42439","1085923","35375087","1424816391","69341765394","4002754789604","269922022642645","20994481541518549","1863251114133428473","186943969920987753459"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A008277","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","0"],["1","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0"],["0","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162435,"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-56658731411f5b0e"]},{"id":162436,"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-56658731411f5b0e"]},{"id":162437,"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-56658731411f5b0e"]}],"region_exact":{},"incoming_edges":[{"id":"e-56658731411f5b0e","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-946a37787f674849","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":"837ae19933fa2c878e84c0ecbcd0c5b46c6b9a57ff7296be3086f3815b58f23e","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-4d60c3dd8627c01d","role":"sequence","name":"Triangle read by rows: T(n,k) = |s(n,n+1-k)|, where s(n,k) are the signed Stirling numbers of the first kind A008276 (1 <= k <= n; in other words, the unsigned Stirling numbers of the first kind in reverse order).","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,         1;\n  2:       6,         9,          2;\n  3:      24,        81,         47,           6;\n  4:     120,       825,        935,         290,          24;\n  5:     720,      9675,      19085,       10765,        2074,         120;\n  6:    5040,    130095,     420665,      383635,      128884,       16884,        720;\n  7:   40320,   1986705,   10165295,    14070070,     7325269,     1628200,     154188,      5040;\n  8:  362880,  34096545,  270026015,   545634670,   412896589,   138383952,   21814316,   1561104,    40320;\n  9: 3628800, 651100275, 7867597325, 22618716925, 23903955124, 11327186661, 2641571792, 310289436, 17356176, 362880;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (n+1+n*x)*P(n-1,x) + n*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(n + 1 + n*x)*P[n - 1, x] + n*x*D[P[n - 1, x], 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] = (n + 1 + n*x)*v[i - 1] + n*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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-4d60c3dd8627c01d","steps":["basis A008277"],"edges":["e-56658731411f5b0e"],"text":"A094638 \u2192 basis A008277"},"headline":"A094638 \u2192 basis A008277","chain":null}