{"id":"n-57fc439b8bd33ef7","revision":0,"first_index":0,"fingerprint":"f6-e22d44c4b2712e20","name":"theta_plus_one of basis_map_A013611 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","8"],["6","56","96"],["24","368","1392","1536"],["120","2608","15792","37376","30720"],["720","20448","172992","645376","1118720","737280"],["5040","177696","1947072","10053376","27192320","37380096","20643840"],["40320","1703808","23039808","153117696","569406720","1212702720","1386233856","660602880"],["362880","17914752","289141056","2361091072","11249368320","32779542528","57750339584","56641978368","23781703680"],["3628800","205274880","3858807168","37489681408","218742781440","813768136704","1954244182016","2942433755136","2531817160704","951268147200"],["39916800","2548327680","54763371648","618189544448","4280654666240","19451103012864","59472532381696","121703648198656","160259507748864","122989156761600","41855798476800"],["479001600","34092610560","825350086656","10631059003392","85368280939520","459431802531840","1712160343240704","4451062549577728","7947444678819840","9310779148861440","6454944768983040","2009078326886400"],["6227020800","489188090880","13184219086848","191026172590080","1747651192417280","10889826415030272","47986265403912192","151788037688000512","343674158501855232","544903845138923520","575523421056663552","364122398879907840","104472072998092800"],["87178291200","7496243435520","222735738304512","3588472272949248","36883817912197120","261511004217262080","1332457184833273856","4976787728778788864","13691038423774003200","27485382989829898240","39225827836739715072","37745587651533078528","21974837492208107520","5850436087893196800"],["1307674368000","122207620147200","3970720523151360","70458019220975616","804450327789404160","6401257626958577664","37072243381360230400","159929075761011359744","518903203269673746432","1264167580104497561600","2281487009724617457664","2962525297162913316864","2619521546576128376832","1413008279310025359360","351026165273591808000"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","10","158","3320","86616","2695536","97399440","4006847808","184871442240","9453831555456","530687942407296","32441760880542720","2145312383081591808","152584569009152501760","11614920660773605840896"],"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","8"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0","0"],["0","0","-16"],["0","8","32"],["-1","-8","-16"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-1","0"],["0","-8"],["2","8"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":190761,"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-4431ead073a9dd8d"]},{"id":190762,"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-4431ead073a9dd8d"]},{"id":190763,"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-4431ead073a9dd8d"]},{"id":190764,"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-4431ead073a9dd8d"]}],"region_exact":{},"incoming_edges":[{"id":"e-4431ead073a9dd8d","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-57fc439b8bd33ef7","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":"d6c60cb3109609cad16905ecb0f61794d2e02b7bfd684ff058c552167d38e758","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-e63fe40b997d2bc1","role":"sequence","name":"basis_map_A013611 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,         8;\n  2:       6,        56,         96;\n  3:      24,       368,       1392,        1536;\n  4:     120,      2608,      15792,       37376,        30720;\n  5:     720,     20448,     172992,      645376,      1118720,       737280;\n  6:    5040,    177696,    1947072,    10053376,     27192320,     37380096,      20643840;\n  7:   40320,   1703808,   23039808,   153117696,    569406720,   1212702720,    1386233856,     660602880;\n  8:  362880,  17914752,  289141056,  2361091072,  11249368320,  32779542528,   57750339584,   56641978368,   23781703680;\n  9: 3628800, 205274880, 3858807168, 37489681408, 218742781440, 813768136704, 1954244182016, 2942433755136, 2531817160704, 951268147200;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = -(n^3+(8*n^3-8*n^2)*x+(16*n^3-32*n^2+16*n)*x^2)*P(n-2,x) + (2*n^2-1+(8*n^2-8*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 8*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 8*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + (8*n^3 - 8*n^2)*x + (16*n^3 - 32*n^2 + 16*n)*x^2)*P[n - 2, x] + (2*n^2 - 1 + (8*n^2 - 8*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 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 + (8*n^3 - 8*n^2)*x + (16*n^3 - 32*n^2 + 16*n)*x^2)*v[i - 2] + (2*n^2 - 1 + (8*n^2 - 8*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 A013611","\u03b8+1"],"edges":["e-60ae4909e1eb1035","e-4431ead073a9dd8d"],"text":"A094638 \u2192 basis A013611 \u2192 \u03b8+1"},"headline":"A094638 \u2192 basis A013611 \u2192 \u03b8+1","chain":null}