{"id":"n-491402ace602b8ce","revision":1,"first_index":0,"fingerprint":"f6-df87a7a77830807f","name":"theta_plus_one of Triangle formed by multiplying Stirling numbers of 2nd kind S2(n,m) (A008277) by m+1.","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":[["2"],["2","6"],["2","18","12"],["2","42","72","20"],["2","90","300","200","30"],["2","186","1080","1300","450","42"],["2","378","3612","7000","4200","882","56"],["2","762","11592","34020","31500","11172","1568","72"],["2","1530","36300","155400","208530","111132","25872","2592","90"],["2","3066","111960","682100","1275750","958734","329280","54000","4050","110"],["2","6138","342012","2915000","7401900","7538454","3583272","855360","103950","6050","132"],["2","12282","1038312","12230020","41382000","55593384","35134176","11449944","2004750","187550","8712","156"],["2","24570","3139500","50650600","225255030","391495104","320063744","136772064","32355180","4325750","320892","12168","182"],["2","49146","9467640","207834900","1202251050","2664327666","2762439680","1505687040","462161700","82802720","8720712","525252","16562","210"],["2","98298","28501212","847119000","6323007600","17669117466","22889514648","15597204480","6041564100","1392891500","195291096","16609320","828100","22050","240"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["2","8","32","136","622","3060","16130","90688","541448","3419052","22752270","159041288","1164414786","8906284280","71001769112"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["2"],["2","6"],["2","18","12"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","3","-6","1"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["0","2","-3"]]},"deriv_order":1,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["0","0","1","-1"]]},"deriv_order":2,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["1","6","-3"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","2"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","1"]]},"deriv_order":2,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","3"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":165171,"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-d901b9327c026219"]},{"id":165172,"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-d901b9327c026219"]},{"id":165173,"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-d901b9327c026219"]}],"region_exact":{},"incoming_edges":[{"id":"e-d901b9327c026219","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-491402ace602b8ce","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":"3bab57a26d5ac83c4d3ef70e35f5bb526628bafc2460fdb6a99ba99a8d8639f1","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-0cecde7ff045a546","role":"sequence","name":"Triangle formed by multiplying Stirling numbers of 2nd kind S2(n,m) (A008277) by m+1.","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: 2;\n  1: 2,    6;\n  2: 2,   18,     12;\n  3: 2,   42,     72,     20;\n  4: 2,   90,    300,    200,      30;\n  5: 2,  186,   1080,   1300,     450,     42;\n  6: 2,  378,   3612,   7000,    4200,    882,     56;\n  7: 2,  762,  11592,  34020,   31500,  11172,   1568,    72;\n  8: 2, 1530,  36300, 155400,  208530, 111132,  25872,  2592,   90;\n  9: 2, 3066, 111960, 682100, 1275750, 958734, 329280, 54000, 4050, 110;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (3*x-6*x^2+x^3)*P(n-3,x) + (2*x-3*x^2)*d/dx P(n-3,x) + (x^2-x^3)*d^2/dx^2 P(n-3,x) + (1+6*x-3*x^2)*P(n-2,x) + 2*x*d/dx P(n-2,x) + x^2*d^2/dx^2 P(n-2,x) + 3*x*P(n-1,x) for n >= 3, with P(0,x) = 2, P(1,x) = 2 + 6*x, P(2,x) = 2 + 18*x + 12*x^2.","mathematica":"P[0, x_] := 2;\nP[1, x_] := 2 + 6*x;\nP[2, x_] := 2 + 18*x + 12*x^2;\nP[n_, x_] := P[n, x] = Expand[(3*x - 6*x^2 + x^3)*P[n - 3, x] + (2*x - 3*x^2)*D[P[n - 3, x], x] + (x^2 - x^3)*D[P[n - 3, x], {x, 2}] + (1 + 6*x - 3*x^2)*P[n - 2, x] + 2*x*D[P[n - 2, x], x] + x^2*D[P[n - 2, x], {x, 2}] + 3*x*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 2; v[2] = 2 + 6*x; v[3] = 2 + 18*x + 12*x^2; for(n = 3, N, my(i = n + 1); v[i] = (3*x - 6*x^2 + x^3)*v[i - 3] + (2*x - 3*x^2)*deriv(v[i - 3], x) + (x^2 - x^3)*deriv(deriv(v[i - 3], x), x) + (1 + 6*x - 3*x^2)*v[i - 2] + 2*x*deriv(v[i - 2], x) + x^2*deriv(deriv(v[i - 2], x), x) + 3*x*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-3cb22164e2f83fe1","steps":["\u03b8+2 (shift +1)","\u03b8+1"],"edges":["e-4ce9900b2631a004","e-d901b9327c026219"],"text":"A008277 \u2192 \u03b8+2 (shift +1) \u2192 \u03b8+1"},"headline":"A008277 \u2192 \u03b8+2 (shift +1) \u2192 \u03b8+1","chain":null}