{"id":"n-f545432d58040336","revision":0,"first_index":0,"fingerprint":"f6-a32349ec41bc229b","name":"theta_plus_one_squared of Triangle T(n,k) read by rows: associated Stirling numbers of first kind (n >= 2, 1 <= k <= floor(n/2)).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["2"],["6","12"],["24","80"],["120","520","135"],["720","3696","1890"],["5040","29232","21420","1680"],["40320","256896","237888","40320"],["362880","2493504","2732940","705600","23625"],["3628800","26547840","33109560","11285120","866250"],["39916800","307992960","425919384","177580480","21656250","374220"],["479001600","3870097920","5827824288","2837303040","471471000","19459440"],["6227020800","52386946560","84770509824","46728409728","9736226500","658377720","6621615"],["87178291200","760241341440","1308695222016","799245807360","198373175000","18722143440","463513050"],["1307674368000","11777241369600","21398487776640","14249234039040","4070794728000","491134644000","20394574200","129729600"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","2","18","104","775","6306","57372","575424","6318549","75437570","973440094","13505157288","200514112747","3172919493506","53315091229080"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one_squared","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["2"],["6","12"],["24","80"]],"recurrence":{"denominator":{"coeffs":[["0"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0","0"],["0","0","-9"],["0","0","0"],["0","0","9"]]},"deriv_order":0,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["0","0","0","0"],["0","0","0","-4"],["0","0","0","0"],["0","0","0","4"]]},"deriv_order":1,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["0","0"],["0","4"],["0","0"],["0","-4"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["0","0","0"],["0","0","-4"],["0","0","0"],["0","0","4"]]},"deriv_order":1,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["0","0"],["1","4"],["0","4"],["-1","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","0"],["0","0","-4"],["0","0","-4"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-1"],["1"],["2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":181466,"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-aa98b96a19f33962"]},{"id":181467,"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-aa98b96a19f33962"]},{"id":181468,"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-aa98b96a19f33962"]}],"region_exact":{},"incoming_edges":[{"id":"e-aa98b96a19f33962","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-f545432d58040336","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":"b05687441583a22fa954a5c365a19dc5050788730165c61f284db603eb1e6e05","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-f13c6b611f4ce3bb","role":"sequence","name":"Triangle T(n,k) read by rows: associated Stirling numbers of first kind (n >= 2, 1 <= k <= floor(n/2)).","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:       1;\n  1:       2;\n  2:       6,       12;\n  3:      24,       80;\n  4:     120,      520,      135;\n  5:     720,     3696,     1890;\n  6:    5040,    29232,    21420,     1680;\n  7:   40320,   256896,   237888,    40320;\n  8:  362880,  2493504,  2732940,   705600,  23625;\n  9: 3628800, 26547840, 33109560, 11285120, 866250;\n","recurrence_sentence":"The row polynomials satisfy n*P(n,x) = (9*n^3-9*n)*x^2*P(n-4,x) + (4*n^3-4*n)*x^3*d/dx P(n-4,x) - (4*n^3-4*n)*x*P(n-3,x) + (4*n^3-4*n)*x^2*d/dx P(n-3,x) - (n^3-n-(4*n^2+4*n)*x)*P(n-2,x) - (4*n^2+4*n)*x^2*d/dx P(n-2,x) + (2*n^2+n-1)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2, P(2,x) = 6 + 12*x, P(3,x) = 24 + 80*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2;\nP[2, x_] := 6 + 12*x;\nP[3, x_] := 24 + 80*x;\nP[n_, x_] := P[n, x] = Expand[Together[((9*n^3 - 9*n)*x^2*P[n - 4, x] + (4*n^3 - 4*n)*x^3*D[P[n - 4, x], x] - (4*n^3 - 4*n)*x*P[n - 3, x] + (4*n^3 - 4*n)*x^2*D[P[n - 3, x], x] - (n^3 - n - (4*n^2 + 4*n)*x)*P[n - 2, x] - (4*n^2 + 4*n)*x^2*D[P[n - 2, x], x] + (2*n^2 + n - 1)*P[n - 1, x])/(n)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2; v[3] = 6 + 12*x; v[4] = 24 + 80*x; for(n = 4, N, my(i = n + 1); v[i] = ((9*n^3 - 9*n)*x^2*v[i - 4] + (4*n^3 - 4*n)*x^3*deriv(v[i - 4], x) - (4*n^3 - 4*n)*x*v[i - 3] + (4*n^3 - 4*n)*x^2*deriv(v[i - 3], x) - (n^3 - n - (4*n^2 + 4*n)*x)*v[i - 2] - (4*n^2 + 4*n)*x^2*deriv(v[i - 2], x) + (2*n^2 + n - 1)*v[i - 1])/(n)); 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-f13c6b611f4ce3bb","steps":["(\u03b8+1)\u00b2"],"edges":["e-aa98b96a19f33962"],"text":"A008306 \u2192 (\u03b8+1)\u00b2"},"headline":"A008306 \u2192 (\u03b8+1)\u00b2","chain":null}