{"id":"n-138d531c6c30f8eb","revision":0,"first_index":0,"fingerprint":"f6-9051f02b0b7c164d","name":"theta_plus_one of veronese_section_2_1 of Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[[],["1"],["1"],["2","2"],["6","12"],["24","70","3"],["120","450","45"],["720","3248","525","4"],["5040","26264","5880","112"],["40320","236248","67347","2184","5"],["362880","2345400","807975","37800","225"],["3628800","25507152","10250790","631092","6600","6"],["39916800","301835952","137987190","10550232","163350","396"],["479001600","3863119104","1971620508","179960924","3747315","16302","7"],["6227020800","53193434112","29873111268","3163772612","83348265","546546","637"]],"degrees":[-1,0,0,1,1,2,2,3,3,4,4,5,5,6,6],"row_sums":["0","1","1","4","18","97","615","4497","37296","346104","3554280","40024440","490453920","6497465760","92541234240"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["1"],["1"],["2","2"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-256","32","-1"],["256","-16","0"],["-96","2","0"],["16","0","0"],["-1","0","0"]]},"deriv_order":0,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["-175","14"],["148","-4"],["-42","0"],["4","0"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["-55","2"],["36","0"],["-6","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-10"],["4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":175633,"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-385be5620dcfdf81"]},{"id":175634,"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-385be5620dcfdf81"]},{"id":175635,"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-385be5620dcfdf81"]}],"region_exact":{},"incoming_edges":[{"id":"e-385be5620dcfdf81","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-138d531c6c30f8eb","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":"4bc8f52af46e835590589540b012a12399fd484352b49fd12f5bfa5f340d50c4","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-b9bba3062ead0647","role":"sequence","name":"veronese_section_2_1 of Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,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":[{"id":"l-d8d7f05668609b5b","kind":"fingerprint_related","node_a_id":"n-138d531c6c30f8eb","node_b_id":"n-a0ecb2e1039e1722","fingerprint":"f6-9051f02b0b7c164d","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-9051f02b0b7c164d"},"other_id":"n-a0ecb2e1039e1722","other":{"id":"n-a0ecb2e1039e1722","name":"theta_plus_one of veronese_section_2_0 of Triangle T(n,k), 0 <= k <= n, read by rows, giving coefficients of the polynomial (x+1)(x+2)...(x+n), expanded in increasing powers of x. T(n,k) is also the unsigned Stirling number |s(n+1, k+1)|, denoting the number of permutations on n+1 elements that contain exactly k+1 cycles.","first_index":0,"fingerprint":"f6-9051f02b0b7c164d","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["2","2"]],"card_rows":[["1"],["1"],["2","2"],["6","12"]],"row_sums_preview":["1","1","4","18","97","615","4497","37296"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-efe23a76953cf69d","steps":["Veronese 2","\u03b8+1"],"edges":["e-4d965342a2d26168","e-bccdee46f84c92d0"],"text":"A130534 \u2192 Veronese 2 \u2192 \u03b8+1"},"headline":"A130534 \u2192 Veronese 2 \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:     0;\n  1:     1;\n  2:     1;\n  3:     2,      2;\n  4:     6,     12;\n  5:    24,     70,     3;\n  6:   120,    450,    45;\n  7:   720,   3248,   525,    4;\n  8:  5040,  26264,  5880,  112;\n  9: 40320, 236248, 67347, 2184, 5;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^4-16*n^3+96*n^2-256*n+256-(2*n^2-16*n+32)*x+x^2)*P(n-4,x) + (4*n^3-42*n^2+148*n-175-(4*n-14)*x)*P(n-3,x) - (6*n^2-36*n+55-2*x)*P(n-2,x) + (4*n-10)*P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 1, P(2,x) = 1, P(3,x) = 2 + 2*x.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[2, x_] := 1;\nP[3, x_] := 2 + 2*x;\nP[n_, x_] := P[n, x] = Expand[-(n^4 - 16*n^3 + 96*n^2 - 256*n + 256 - (2*n^2 - 16*n + 32)*x + x^2)*P[n - 4, x] + (4*n^3 - 42*n^2 + 148*n - 175 - (4*n - 14)*x)*P[n - 3, x] - (6*n^2 - 36*n + 55 - 2*x)*P[n - 2, x] + (4*n - 10)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; v[3] = 1; v[4] = 2 + 2*x; for(n = 4, N, my(i = n + 1); v[i] = -(n^4 - 16*n^3 + 96*n^2 - 256*n + 256 - (2*n^2 - 16*n + 32)*x + x^2)*v[i - 4] + (4*n^3 - 42*n^2 + 148*n - 175 - (4*n - 14)*x)*v[i - 3] - (6*n^2 - 36*n + 55 - 2*x)*v[i - 2] + (4*n - 10)*v[i - 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-b9fbfb57d9269e2f","steps":["Veronese 2 (j=1)","\u03b8+1"],"edges":["e-b4f12c588dbcf356","e-385be5620dcfdf81"],"text":"A132393 \u2192 Veronese 2 (j=1) \u2192 \u03b8+1"},"headline":"A132393 \u2192 Veronese 2 (j=1) \u2192 \u03b8+1","chain":null}