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Stirling set numbers of order 2, T(n, n) = 1, T(n, k) = 0 if k < 0 or k > n, otherwise T(n, k) = T(n-1, k-1) + k^2*T(n-1, k), for 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","9"],["0","4","45","16"],["0","4","189","224","25"],["0","4","765","2352","750","36"],["0","4","3069","22528","15675","1980","49"],["0","4","12285","208208","286000","72072","4459","64"],["0","4","49149","1895712","4901325","2213640","258622","8960","81"],["0","4","196605","17148784","81383250","62398908","12323402","776832","16524","100"],["0","4","786429","154688576","1328926975","1677164580","528574319","54160640","2040714","28500","121"],["0","4","3145725","1393595280","21504532500","43842769344","21311482829","3344254144","199152756","4827900","46585","144"],["0","4","12582909","12547949920","346250012625","1127035760400","826888262340","191703859200","16978348035","636927500","10500259","72864","169"],["0","4","50331645","112953918896","5559606373750","28674494028180","31301998340340","10473506423040","1329239471040","72552051000","1820708175","21312720","109850","196"],["0","4","201326589","1016674748544","89130192478275","724868183882700","1165901112679485","554086057459200","98326857713220","7517752515000","269858799210","4745632320","40831245","160524","225"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","13","65","442","3907","43305","583092","9327493","174244409","3746370858","91603807211","2522064276225","77526243068836","2641121678226541"],"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"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","4","9"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","1","5","4"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","5","9"]]},"deriv_order":2,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","0","2","4"]]},"deriv_order":3,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-1","-4"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","1"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":178377,"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-f9f7328fa18225f7"]},{"id":178378,"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-f9f7328fa18225f7"]},{"id":178379,"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-f9f7328fa18225f7"]}],"region_exact":{},"incoming_edges":[{"id":"e-f9f7328fa18225f7","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-8f9c04795daca07f","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":"48c9808e00135c3ee38ff663d4682b2d3b30c633227d6eb8416fcfbc770fec9a","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-0164479e47952770","role":"sequence","name":"Triangle read by rows. Stirling set numbers of order 2, T(n, n) = 1, T(n, k) = 0 if k < 0 or k > n, otherwise T(n, k) = T(n-1, k-1) + k^2*T(n-1, k), for 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,      9;\n   4: 0, 4,     45,       16;\n   5: 0, 4,    189,      224,       25;\n   6: 0, 4,    765,     2352,      750,       36;\n   7: 0, 4,   3069,    22528,    15675,     1980,       49;\n   8: 0, 4,  12285,   208208,   286000,    72072,     4459,     64;\n   9: 0, 4,  49149,  1895712,  4901325,  2213640,   258622,   8960,    81;\n  10: 0, 4, 196605, 17148784, 81383250, 62398908, 12323402, 776832, 16524, 100;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (4*x+9*x^2)*P(n-2,x) + (x+5*x^2+4*x^3)*d/dx P(n-2,x) + (5*x^2+9*x^3)*d^2/dx^2 P(n-2,x) + (2*x^3+4*x^4)*d^3/dx^3 P(n-2,x) + (1+4*x)*P(n-1,x) - (x+4*x^2)*d/dx P(n-1,x) + x^2*d^2/dx^2 P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 4*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 4*x;\nP[n_, x_] := P[n, x] = Expand[(4*x + 9*x^2)*P[n - 2, x] + (x + 5*x^2 + 4*x^3)*D[P[n - 2, x], x] + (5*x^2 + 9*x^3)*D[P[n - 2, x], {x, 2}] + (2*x^3 + 4*x^4)*D[P[n - 2, x], {x, 3}] + (1 + 4*x)*P[n - 1, x] - (x + 4*x^2)*D[P[n - 1, x], x] + x^2*D[P[n - 1, x], {x, 2}]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 4*x; for(n = 3, N, my(i = n); v[i] = (4*x + 9*x^2)*v[i - 2] + (x + 5*x^2 + 4*x^3)*deriv(v[i - 2], x) + (5*x^2 + 9*x^3)*deriv(deriv(v[i - 2], x), x) + (2*x^3 + 4*x^4)*deriv(deriv(deriv(v[i - 2], x), x), x) + (1 + 4*x)*v[i - 1] - (x + 4*x^2)*deriv(v[i - 1], x) + x^2*deriv(deriv(v[i - 1], x), x)); 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-0164479e47952770","steps":["(\u03b8+1)\u00b2"],"edges":["e-f9f7328fa18225f7"],"text":"A269945 \u2192 (\u03b8+1)\u00b2"},"headline":"A269945 \u2192 (\u03b8+1)\u00b2","chain":null}