{"id":"n-fe015a7293beddbf","revision":0,"first_index":0,"fingerprint":"f6-5cdcbb7c4ab4eee9","name":"theta_plus_one of basis_map_A008277 of Unsigned 2-Stirling numbers of the first kind.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["12","16","3"],["60","120","45","4"],["360","960","540","96","5"],["2520","8400","6300","1680","175","6"],["20160","80640","75600","26880","4200","288","7"],["181440","846720","952560","423360","88200","9072","441","8"],["1814400","9676800","12700800","6773760","1764000","241920","17640","640","9"],["19958400","119750400","179625600","111767040","34927200","5987520","582120","31680","891","10"],["239500800","1596672000","2694384000","1916006400","698544000","143700480","17463600","1267200","53460","1200","11"],["3113510400","22832409600","42810768000","34248614400","14270256000","3424861440","499458960","45302400","2548260","85800","1573","12"],["43589145600","348713164800","719220902400","639307468800","299675376000","82196674560","13984850880","1522160640","107026920","4804800","132132","2016","13"],["653837184000","5666588928000","12749825088000","12466495641600","6492966480000","2003543942400","389577988800","49470220800","4174049880","234234000","8588580","196560","2535","14"],["10461394944000","97639686144000","237996734976000","253863183974400","145442449152000","49865982566400","10908183686400","1583047065600","155831195520","10493683200","480960480","14676480","283920","3136","15"],["177843714048000","1778437140480000","4668397493760000","5394592659456000","3371620412160000","1271582555443200","309065204448000","50459625216000","5676707836800","445981536000","24528984480","935625600","24133200","399840","3825","16"],["3201186852864000","34145993097216000","96035605585920000","119510975840256000","80918889891840000","33292343269785600","8901077888102400","1614708006912000","204361482124800","18348954624000","1177391255040","53892034560","1737590400","38384640","550800","4608","17"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["31","229","1961","19081","207775","2501801","32989969","472630861","7307593151","121247816845","2148321709561","40476722545169","807927483311551","17028146983530961","377844723929464865"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["12","16","3"]],"recurrence":{"denominator":{"coeffs":[["1","2"],["0","1"]]},"terms":[{"coeff":{"coeffs":[["4","10","3"],["1","6","1"],["0","1","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","3"],["0","0","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":188409,"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-321be6c91e39be05"]},{"id":188410,"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-321be6c91e39be05"]},{"id":188411,"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-321be6c91e39be05"]},{"id":188412,"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-321be6c91e39be05"]}],"region_exact":{},"incoming_edges":[{"id":"e-321be6c91e39be05","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-fe015a7293beddbf","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":"8e70c28a4edc32cdfe3979ca7047c21b7c5d50c029d6b5c2c7854c9a98d80214","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-271026fda29dd6cb","role":"sequence","name":"basis_map_A008277 of Unsigned 2-Stirling numbers of the first kind.","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":[{"id":"l-929674a90916da14","kind":"fingerprint_related","node_a_id":"n-4ce499352d70bef2","node_b_id":"n-fe015a7293beddbf","fingerprint":"f6-5cdcbb7c4ab4eee9","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-5cdcbb7c4ab4eee9"},"other_id":"n-4ce499352d70bef2","other":{"id":"n-4ce499352d70bef2","name":"theta_plus_one of basis_map_A008277 of Triangle read by rows: T(n,k) = (-1)^(n+k)*Sum_{j=1..k} s(n,j), where s(n,j) are the signed Stirling numbers of the first kind (n >= 2; 1 <= k <= n-1; s(n,j) = A008275(n,j)).","first_index":0,"fingerprint":"f6-5cdcbb7c4ab4eee9","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","2"],["12","16","3"]],"card_rows":[["1"],["3","2"],["12","16","3"],["60","120","45","4"]],"row_sums_preview":["1","5","31","229","1961","19081","207775","2501801"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-914700b1060feeb2","steps":["basis A008277","\u03b8+1"],"edges":["e-0a932d25d1f1a092","e-2cfefc58c5bc5563"],"text":"A136124 \u2192 basis A008277 \u2192 \u03b8+1"},"headline":"A136124 \u2192 basis A008277 \u2192 \u03b8+1","chain":null}}],"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:         12,          16,           3;\n  1:         60,         120,          45,           4;\n  2:        360,         960,         540,          96,           5;\n  3:       2520,        8400,        6300,        1680,         175,          6;\n  4:      20160,       80640,       75600,       26880,        4200,        288,         7;\n  5:     181440,      846720,      952560,      423360,       88200,       9072,       441,        8;\n  6:    1814400,     9676800,    12700800,     6773760,     1764000,     241920,     17640,      640,       9;\n  7:   19958400,   119750400,   179625600,   111767040,    34927200,    5987520,    582120,    31680,     891,    10;\n  8:  239500800,  1596672000,  2694384000,  1916006400,   698544000,  143700480,  17463600,  1267200,   53460,  1200,   11;\n  9: 3113510400, 22832409600, 42810768000, 34248614400, 14270256000, 3424861440, 499458960, 45302400, 2548260, 85800, 1573, 12;\n","recurrence_sentence":"The row polynomials satisfy (1+(n+2)*x)*P(n,x) = (n+4+(n^2+6*n+10)*x+(n+3)*x^2)*P(n-1,x) + (x+(n+3)*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 12 + 16*x + 3*x^2.","mathematica":"P[0, x_] := 12 + 16*x + 3*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((n + 4 + (n^2 + 6*n + 10)*x + (n + 3)*x^2)*P[n - 1, x] + (x + (n + 3)*x^2)*D[P[n - 1, x], x])/(1 + (n + 2)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 12 + 16*x + 3*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((n + 4 + (n^2 + 6*n + 10)*x + (n + 3)*x^2)*v[i - 1] + (x + (n + 3)*x^2)*deriv(v[i - 1], x))/(1 + (n + 2)*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-c237629cf5fdbb07","steps":["basis A008277","\u03b8+1"],"edges":["e-13b78624414ee2b1","e-321be6c91e39be05"],"text":"A143491 \u2192 basis A008277 \u2192 \u03b8+1"},"headline":"A143491 \u2192 basis A008277 \u2192 \u03b8+1","chain":null}
