{"id":"n-66a033f7e5a78ba8","revision":0,"first_index":0,"fingerprint":"f6-10c290ce97befc40","name":"theta_plus_one of basis_map_A013609 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","4"],["6","28","24"],["24","184","348","192"],["120","1304","3948","4672","1920"],["720","10224","43248","80672","69920","23040"],["5040","88848","486768","1256672","1699520","1168128","322560"],["40320","851904","5759952","19139712","35587920","37896960","21659904","5160960"],["362880","8957376","72285264","295136384","703085520","1024360704","902349056","442515456","92897280"],["3628800","102637440","964701792","4686210176","13671423840","25430254272","30535065344","22987763712","9889910784","1857945600"],["39916800","1274163840","13690842912","77273693056","267540916640","607846969152","929258318464","950809751552","626013702144","240213196800","40874803200"],["479001600","17046305280","206337521664","1328882375424","5335517558720","14357243829120","26752505363136","34773926168576","31044705776640","18185115525120","6303657000960","980995276800"],["6227020800","244594045440","3296054771712","23878271573760","109228199526080","340307075469696","749785396936128","1185844044437504","1342477181647872","1064265322536960","562034590875648","177794140078080","25505877196800"],["87178291200","3748121717760","55683934576128","448559034118656","2305238619512320","8172218881789440","20819643513019904","38881154131084288","53480618842867200","53682388652011520","38306472496816128","18430462720475136","5364950559621120","714164561510400"],["1307674368000","61103810073600","992680130787840","8807252402621952","50278145486837760","200039300842455552","579253802833753600","1249445904382901248","2026965637772163072","2469077304891596800","2228014657934196736","1446545555255328768","639531627582062592","172486362220462080","21424936845312000"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","6","58","748","11964","227824","5027536","126097632","3541949920","110129541760","3754836274560","139286411703040","5584666976116480","240665391247411200","11092925580064921600"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["2","4"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0","0"],["0","0","-4"],["0","4","8"],["-1","-4","-4"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-1","0"],["0","-4"],["2","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":187139,"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-902caecd397dd1cc"]},{"id":187140,"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-902caecd397dd1cc"]},{"id":187141,"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-902caecd397dd1cc"]},{"id":187142,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-902caecd397dd1cc"]}],"region_exact":{},"incoming_edges":[{"id":"e-902caecd397dd1cc","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-66a033f7e5a78ba8","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":"d926b52d0cb1877a97ffa430162e2c2c40c307b28dd0b6e99d6663bb19b95dfc","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-656046abebe1865b","role":"sequence","name":"basis_map_A013609 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).","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,         4;\n  2:       6,        28,        24;\n  3:      24,       184,       348,        192;\n  4:     120,      1304,      3948,       4672,        1920;\n  5:     720,     10224,     43248,      80672,       69920,       23040;\n  6:    5040,     88848,    486768,    1256672,     1699520,     1168128,      322560;\n  7:   40320,    851904,   5759952,   19139712,    35587920,    37896960,    21659904,     5160960;\n  8:  362880,   8957376,  72285264,  295136384,   703085520,  1024360704,   902349056,   442515456,   92897280;\n  9: 3628800, 102637440, 964701792, 4686210176, 13671423840, 25430254272, 30535065344, 22987763712, 9889910784, 1857945600;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = -(n^3+(4*n^3-4*n^2)*x+(4*n^3-8*n^2+4*n)*x^2)*P(n-2,x) + (2*n^2-1+(4*n^2-4*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + (4*n^3 - 4*n^2)*x + (4*n^3 - 8*n^2 + 4*n)*x^2)*P[n - 2, x] + (2*n^2 - 1 + (4*n^2 - 4*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 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 + (4*n^3 - 4*n^2)*x + (4*n^3 - 8*n^2 + 4*n)*x^2)*v[i - 2] + (2*n^2 - 1 + (4*n^2 - 4*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-4d60c3dd8627c01d","steps":["basis A013609","\u03b8+1"],"edges":["e-23134a27b4e4bcc8","e-902caecd397dd1cc"],"text":"A094638 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A094638 \u2192 basis A013609 \u2192 \u03b8+1","chain":null}