{"id":"n-abc0ba18569a86e9","revision":1,"first_index":0,"fingerprint":"f6-9edb451024733e5b","name":"basis_map_A008277 of Triangle read by rows, Stirling cycle numbers of order 2, T(n, n) = 1, T(n, k) = 0 if k < 0 or k > n, otherwise T(n, k) = T(n-1, k-1) + (n-1)^2*T(n-1, k), for 0 <= k <= n.","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"],["1","1"],["2","4","1"],["10","26","11","1"],["100","296","158","24","1"],["1700","5428","3298","638","45","1"],["44200","148256","97772","21800","1988","76","1"],["1635400","5677928","3961364","969772","103308","5180","119","1"],["81770000","291209728","211668856","55359280","6548404","388208","11844","176","1"],["5315050000","19301612048","14473023080","3976099896","507199156","33722964","1229132","24516","249","1"],["435834100000","1607348849984","1235035550768","352441514240","47595227312","3441097024","141886580","3411056","46926","340","1"],["44019244100000","164385416798368","128816010579088","37888953031728","5349940382000","412351511856","18622961084","510280628","8525990","84326","451","1"],["5370347780200000","20263425510299264","16137570728605280","4864935139545088","711981441163728","57718582387712","2796090530608","84449162096","1618659328","19572696","143858","584","1"],["778700428129000000","2963830472283892544","2392486322615275424","736147971381278304","110950169872940560","9369768799320528","479928252509520","15632363169200","332104039280","4632854512","41870686","234962","741","1"],["132379072781930000000","507593711188674625024","414471477962111265472","129746085371576422016","20041477529272935744","1750659709754032960","93837141240996048","3246856533457920","74746882161040","1161384996928","12169577992","84398808","369824","924","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","2","7","48","579","11110","314093","12353072","646956497","43607961042","3681841684231","380891058255520","47408861360270243","6991981030476386262","1206079644461615240701"],"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":[["2","1"],["-2","0"],["1","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":166067,"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-0687c4d1ba7aff91"]},{"id":166068,"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-0687c4d1ba7aff91"]},{"id":166069,"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-0687c4d1ba7aff91"]}],"region_exact":{},"incoming_edges":[{"id":"e-0687c4d1ba7aff91","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-abc0ba18569a86e9","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":"3fe11f77cdd1f42a1be28970654ad3f764c9ec26f3f4fc6a7049fda7a979b971","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-54284d24a094e2d3","role":"sequence","name":"Triangle read by rows, Stirling cycle numbers of order 2, T(n, n) = 1, T(n, k) = 0 if k < 0 or k > n, otherwise T(n, k) = T(n-1, k-1) + (n-1)^2*T(n-1, k), for 0 <= k <= n.","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:          1,           1;\n  2:          2,           4,           1;\n  3:         10,          26,          11,          1;\n  4:        100,         296,         158,         24,         1;\n  5:       1700,        5428,        3298,        638,        45,        1;\n  6:      44200,      148256,       97772,      21800,      1988,       76,       1;\n  7:    1635400,     5677928,     3961364,     969772,    103308,     5180,     119,     1;\n  8:   81770000,   291209728,   211668856,   55359280,   6548404,   388208,   11844,   176,   1;\n  9: 5315050000, 19301612048, 14473023080, 3976099896, 507199156, 33722964, 1229132, 24516, 249, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (n^2-2*n+2+x)*P(n-1,x) + 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^2 - 2*n + 2 + x)*P[n - 1, x] + 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^2 - 2*n + 2 + x)*v[i - 1] + 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-54284d24a094e2d3","steps":["basis A008277"],"edges":["e-0687c4d1ba7aff91"],"text":"A269944 \u2192 basis A008277"},"headline":"A269944 \u2192 basis A008277","chain":null}