{"id":"n-ab6e8549e7f75941","revision":0,"first_index":0,"fingerprint":"f6-a25633ab4d4ed031","name":"theta_plus_one_squared of Triangle read by rows, Stirling cycle 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) + (n-1)^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":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["0","4"],["0","4","9"],["0","16","45","16"],["0","144","441","224","25"],["0","2304","7380","4368","750","36"],["0","57600","189684","122320","25575","1980","49"],["0","2073600","6958224","4740736","1111825","108108","4459","64"],["0","101606400","345618576","244666240","61886825","6898320","365638","8960","81"],["0","6502809600","22348203264","16273072384","4343047800","530609508","32790212","1051008","16524","100"],["0","526727577600","1824835785984","1357849002240","377213547400","49233358980","3378225669","127959680","2668626","28500","121"],["0","52672757760000","183668715648000","139029052732416","39842993806000","5466523406256","404834638845","17208344384","428811570","6144600","46585","144"],["0","6373403688960000","22342428298368000","17149037541774336","5038235145420400","718823243237616","56425537047649","2610973280384","73665510831","1272893600","13071751","72864","169"],["0","917770131210240000","3231649833265152000","2509181278545936384","752301232099560000","110765605635622080","9103675638157100","449678813008960","13912346617650","274241753500","3422533400","26048880","109850","196"],["0","155103152174530560000","548213804617033728000","429796791333401296896","131059503972553665600","19802701126643497920","1689285479408146620","87886234517739840","2920311326097315","63522592906500","910240666335","8475358320","49135905","160524","225"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","4","13","77","834","14838","397208","14997016","761051040","50031600400","4139368154800","421102521338800","51681039379637600","7531235625244740000","1285756109690772960000"],"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"],["0","4"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-16","-16","9"],["32","16","0"],["-24","-4","0"],["8","0","0"],["-1","0","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","16","4"],["0","0","-16","0"],["0","0","4","0"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["5","4"],["-6","0"],["2","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":185960,"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-d8635e81b8515cd4"]},{"id":185961,"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-d8635e81b8515cd4"]},{"id":185962,"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-d8635e81b8515cd4"]}],"region_exact":{},"incoming_edges":[{"id":"e-d8635e81b8515cd4","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-ab6e8549e7f75941","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":"c14f8227af4bdbf2db8bb2b178f9abc8daf74b1715a7e576615b662c7d5ee092","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-54284d24a094e2d3","role":"sequence","name":"Triangle read by rows, Stirling cycle 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) + (n-1)^2*T(n-1, 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":[],"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: 0,          4;\n  2: 0,          4,           9;\n  3: 0,         16,          45,          16;\n  4: 0,        144,         441,         224,         25;\n  5: 0,       2304,        7380,        4368,        750,        36;\n  6: 0,      57600,      189684,      122320,      25575,      1980,       49;\n  7: 0,    2073600,     6958224,     4740736,    1111825,    108108,     4459,      64;\n  8: 0,  101606400,   345618576,   244666240,   61886825,   6898320,   365638,    8960,    81;\n  9: 0, 6502809600, 22348203264, 16273072384, 4343047800, 530609508, 32790212, 1051008, 16524, 100;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^4-8*n^3+24*n^2-32*n+16+(4*n^2-16*n+16)*x-9*x^2)*P(n-2,x) + ((4*n^2-16*n+16)*x^2+4*x^3)*d/dx P(n-2,x) + (2*n^2-6*n+5+4*x)*P(n-1,x) - 4*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4*x;\nP[n_, x_] := P[n, x] = Expand[-(n^4 - 8*n^3 + 24*n^2 - 32*n + 16 + (4*n^2 - 16*n + 16)*x - 9*x^2)*P[n - 2, x] + ((4*n^2 - 16*n + 16)*x^2 + 4*x^3)*D[P[n - 2, x], x] + (2*n^2 - 6*n + 5 + 4*x)*P[n - 1, x] - 4*x^2*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^4 - 8*n^3 + 24*n^2 - 32*n + 16 + (4*n^2 - 16*n + 16)*x - 9*x^2)*v[i - 2] + ((4*n^2 - 16*n + 16)*x^2 + 4*x^3)*deriv(v[i - 2], x) + (2*n^2 - 6*n + 5 + 4*x)*v[i - 1] - 4*x^2*deriv(v[i - 1], x)); 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-54284d24a094e2d3","steps":["(\u03b8+1)\u00b2"],"edges":["e-d8635e81b8515cd4"],"text":"A269944 \u2192 (\u03b8+1)\u00b2"},"headline":"A269944 \u2192 (\u03b8+1)\u00b2","chain":null}