{"id":"n-35cc86536da6692a","revision":0,"first_index":1,"fingerprint":"f6-be569a6fd03aad81","name":"theta_plus_one_squared of Triangle read by rows, Ward numbers T(n, k) = Sum_{m=0..k} (-1)^(m + k) * binomial(n + k, n + m) * Stirling2(n + m, m), for n >= 0, 0 <= k <= n.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["0","4"],["0","4","27"],["0","4","90","240"],["0","4","225","1680","2625"],["0","4","504","7840","31500","34020"],["0","4","1071","30688","236250","623700","509355"],["0","4","2214","109200","1424500","6860700","13243230","8648640"],["0","4","4509","366960","7574875","58918860","200855655","302702400","164189025"],["0","4","9108","1189056","37180000","436395960","2327865540","6054048000","7443235800","3445942500"],["0","4","18315","3761472","172872700","2931528600","22876944090","91025975040","189802512900","196418722500","79222218075"],["0","4","36738","11707696","773773000","18391693320","201103842940","1145143359360","3592105597080","6219926212500","5545555265250","1979900722800"],["0","4","73593","36037456","3369491125","109786196520","1632186405850","12743995904640","56273963996250","144673301272500","213503877712125","166971627622800","53443570205025"],["0","4","147312","110074848","14378900900","631416125340","12482880249840","129712842138880","772774108306200","2761014333577500","5986242056995200","7680694870648800","5344357020502500","1549547301802500"],["0","4","294759","334414752","60439466750","3529781738028","91226404073805","1234073297416960","9629716998922020","45838055400637500","136701449473413825","255466590262884000","289486005277218750","181297034310892500","48028060502296875"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","31","334","4534","73868","1401068","30288488","734612288","19745865968","582454553696","18702912210688","649355714917888","24237471319469824","967775802483670528"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one_squared","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["0","4"],["0","4","27"]],"recurrence":{"denominator":{"coeffs":[["9"],["-6"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","8","288"],["0","-4","-498"],["0","0","317"],["0","0","-88"],["0","0","9"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","-1","-26","-81"],["0","0","19","36"],["0","0","-4","39"],["0","0","0","-26"],["0","0","0","4"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","-7","-70","-63"],["0","0","5","83","78"],["0","0","-1","-32","-31"],["0","0","0","4","4"]]},"deriv_order":2,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-6","-36"],["5","60"],["-1","-28"],["0","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","16","72"],["0","-11","-84"],["0","2","32"],["0","0","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":183767,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-3122244141ab1bdb"]},{"id":183768,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-3122244141ab1bdb"]},{"id":183769,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-3122244141ab1bdb"]}],"region_exact":{},"incoming_edges":[{"id":"e-3122244141ab1bdb","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"2\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-35cc86536da6692a","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":"231df6628f807e08654ddefcdc8bd89d2a550e5e439044c5354ac7456beb638b","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-eda3c556a6b4857f","role":"sequence","name":"Triangle read by rows, Ward numbers T(n, k) = Sum_{m=0..k} (-1)^(m + k) * binomial(n + k, n + m) * Stirling2(n + m, m), for n >= 0, 0 <= k <= n.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1;\n   2: 0, 4;\n   3: 0, 4,   27;\n   4: 0, 4,   90,     240;\n   5: 0, 4,  225,    1680,     2625;\n   6: 0, 4,  504,    7840,    31500,     34020;\n   7: 0, 4, 1071,   30688,   236250,    623700,     509355;\n   8: 0, 4, 2214,  109200,  1424500,   6860700,   13243230,    8648640;\n   9: 0, 4, 4509,  366960,  7574875,  58918860,  200855655,  302702400,  164189025;\n  10: 0, 4, 9108, 1189056, 37180000, 436395960, 2327865540, 6054048000, 7443235800, 3445942500;\n","recurrence_sentence":"The row polynomials satisfy (n^2-6*n+9)*P(n,x) = -((4*n-8)*x-(9*n^4-88*n^3+317*n^2-498*n+288)*x^2)*P(n-2,x) - (x+(4*n^2-19*n+26)*x^2-(4*n^4-26*n^3+39*n^2+36*n-81)*x^3)*d/dx P(n-2,x) - ((n^2-5*n+7)*x^2-(4*n^3-32*n^2+83*n-70)*x^3-(4*n^3-31*n^2+78*n-63)*x^4)*d^2/dx^2 P(n-2,x) - (n^2-5*n+6-(4*n^3-28*n^2+60*n-36)*x)*P(n-1,x) + ((2*n^2-11*n+16)*x-(4*n^3-32*n^2+84*n-72)*x^2)*d/dx P(n-1,x) for n >= 4, with P(1,x) = 1, P(2,x) = 4*x, P(3,x) = 4*x + 27*x^2.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 4*x;\nP[3, x_] := 4*x + 27*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[(-((4*n - 8)*x - (9*n^4 - 88*n^3 + 317*n^2 - 498*n + 288)*x^2)*P[n - 2, x] - (x + (4*n^2 - 19*n + 26)*x^2 - (4*n^4 - 26*n^3 + 39*n^2 + 36*n - 81)*x^3)*D[P[n - 2, x], x] - ((n^2 - 5*n + 7)*x^2 - (4*n^3 - 32*n^2 + 83*n - 70)*x^3 - (4*n^3 - 31*n^2 + 78*n - 63)*x^4)*D[P[n - 2, x], {x, 2}] - (n^2 - 5*n + 6 - (4*n^3 - 28*n^2 + 60*n - 36)*x)*P[n - 1, x] + ((2*n^2 - 11*n + 16)*x - (4*n^3 - 32*n^2 + 84*n - 72)*x^2)*D[P[n - 1, x], x])/(n^2 - 6*n + 9)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 4*x; v[3] = 4*x + 27*x^2; for(n = 4, N, my(i = n); v[i] = (-((4*n - 8)*x - (9*n^4 - 88*n^3 + 317*n^2 - 498*n + 288)*x^2)*v[i - 2] - (x + (4*n^2 - 19*n + 26)*x^2 - (4*n^4 - 26*n^3 + 39*n^2 + 36*n - 81)*x^3)*deriv(v[i - 2], x) - ((n^2 - 5*n + 7)*x^2 - (4*n^3 - 32*n^2 + 83*n - 70)*x^3 - (4*n^3 - 31*n^2 + 78*n - 63)*x^4)*deriv(deriv(v[i - 2], x), x) - (n^2 - 5*n + 6 - (4*n^3 - 28*n^2 + 60*n - 36)*x)*v[i - 1] + ((2*n^2 - 11*n + 16)*x - (4*n^3 - 32*n^2 + 84*n - 72)*x^2)*deriv(v[i - 1], x))/(n^2 - 6*n + 9)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-eda3c556a6b4857f","steps":["(\u03b8+1)\u00b2"],"edges":["e-3122244141ab1bdb"],"text":"A269939 \u2192 (\u03b8+1)\u00b2"},"headline":"A269939 \u2192 (\u03b8+1)\u00b2","chain":null}