{"id":"n-92d6e06b89093e78","revision":0,"first_index":0,"fingerprint":"f6-6ea9ebd1c906150f","name":"theta_plus_one 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":false,"rows":[["1"],["0","2"],["0","2","6"],["0","4","21","24"],["0","12","87","184","120"],["0","48","438","1316","1630","720"],["0","240","2622","10084","18020","15336","5040"],["0","1440","18252","84976","196355","243384","155484","40320"],["0","10080","145044","789520","2224245","3588696","3359972","1703808","362880"],["0","80640","1296432","8056688","26676060","52731414","64561084","48190176","20154096","3628800"],["0","725760","12877488","89795952","340793140","794695446","1196249586","1171552544","723526344","256593600","39916800"],["0","7257600","140749920","1086829344","4642625740","12445423908","22161087417","26754091664","21733226172","11409035760","3503950560","479001600"],["0","79833600","1678885920","14207121504","67371323300","203753473644","418007896299","598218492880","600245725356","415275742320","189142728192","51138915840","6227020800"],["0","958003200","21703386240","199586147328","1039321604040","3496034327208","8106355772522","13389024945288","15951894162642","13653524940760","8208155853480","3296054771712","794930647680","87178291200"],["0","12454041600","302262088320","2999749791744","17003938430760","62909049201576","162484519053850","303759016649096","418251767002632","425636400093200","316970310182944","168209637794496","60324262457472","13118426012160","1307674368000"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","2","8","49","403","4152","51342","740211","12184245","225375390","4626726660","104363279685","2565347159655","68244722853300","1953289467167850"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["0","2"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0","0"],["-4","-4","-1"],["4","6","2"],["-1","-2","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1","0"],["-4","-2"],["2","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":180782,"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-172e5644d6aad2bb"]},{"id":180783,"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-172e5644d6aad2bb"]},{"id":180784,"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-172e5644d6aad2bb"]}],"region_exact":{},"incoming_edges":[{"id":"e-172e5644d6aad2bb","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-92d6e06b89093e78","premise_count":1,"theorem":"Laguerre's theorem on multiplier sequences: if f is a real polynomial whose roots are all real and <= 0, then (f(k))_(k >= 0) is a multiplier sequence, so x^k -> f(k) x^k maps every real-rooted polynomial to a real-rooted polynomial or zero (Laguerre; see T. Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"certificate_hash":"2d6ef9b36f2adc37f0ec41902894263ea9f76fb7d42668680af2aa57309cde7e","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-849de7332df28657","role":"sequence","name":"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":[],"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: 0,     2;\n  2: 0,     2,       6;\n  3: 0,     4,      21,      24;\n  4: 0,    12,      87,     184,      120;\n  5: 0,    48,     438,    1316,     1630,      720;\n  6: 0,   240,    2622,   10084,    18020,    15336,     5040;\n  7: 0,  1440,   18252,   84976,   196355,   243384,   155484,    40320;\n  8: 0, 10080,  145044,  789520,  2224245,  3588696,  3359972,  1703808,   362880;\n  9: 0, 80640, 1296432, 8056688, 26676060, 52731414, 64561084, 48190176, 20154096, 3628800;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = -(n^3-4*n^2+4*n+(2*n^3-6*n^2+4*n)*x+(n^3-2*n^2+n)*x^2)*P(n-2,x) + (2*n^2-4*n+1+(2*n^2-2*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^3 - 4*n^2 + 4*n + (2*n^3 - 6*n^2 + 4*n)*x + (n^3 - 2*n^2 + n)*x^2)*P[n - 2, x] + (2*n^2 - 4*n + 1 + (2*n^2 - 2*n)*x)*P[n - 1, x])/(n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 - 4*n^2 + 4*n + (2*n^3 - 6*n^2 + 4*n)*x + (n^3 - 2*n^2 + n)*x^2)*v[i - 2] + (2*n^2 - 4*n + 1 + (2*n^2 - 2*n)*x)*v[i - 1])/(n - 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":["\u03b8+1"],"edges":["e-172e5644d6aad2bb"],"text":"A088996 \u2192 \u03b8+1"},"headline":"A088996 \u2192 \u03b8+1","chain":null}