{"id":"n-0c8274243629c17a","revision":0,"first_index":0,"fingerprint":"f6-53219bd9184fb09a","name":"theta_plus_one_squared of T(n,k) is the number of partial bijections of height k (height(alpha) = |Im(alpha)|) of an n-element set.","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"],["1","4"],["1","16","18"],["1","36","162","96"],["1","64","648","1536","600"],["1","100","1800","9600","15000","4320"],["1","144","4050","38400","135000","155520","35280"],["1","196","7938","117600","735000","1905120","1728720","322560"],["1","256","14112","301056","2940000","13547520","27659520","20643840","3265920"],["1","324","23328","677376","9525600","68584320","248935680","418037760","264539520","36288000"],["1","400","36450","1382400","26460000","274337280","1555848000","4644864000","6613488000","3628800000","439084800"],["1","484","54450","2613600","65340000","922078080","7530304320","35126784000","88914672000","109771200000","53129260800","5748019200"],["1","576","78408","4646400","147015000","2709780480","30121217280","202330275840","800232048000","1756339200000","1912653388800","827714764800","80951270400"],["1","676","109512","7852416","306735000","7155514080","103887463680","949828239360","5409568644480","18551332800000","35915380300800","34970948812800","13680764697600","1220496076800"],["1","784","149058","12719616","601200600","17314577280","318155357520","3799312957440","29452095953280","145442449152000","439963408684800","761589551923200","670357470182400","239217231052800","19615115520000"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","35","295","2849","30821","368395","4817135","68372225","1046611909","17184301331","301210326935","5613203685985","110809677247205","2309772719430779"],"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"]],"recurrence":{"denominator":{"coeffs":[["1","0","0"],["0","-2","0"],["0","3","0"],["0","0","0"],["0","0","1"]]},"terms":[{"coeff":{"coeffs":[["1","-4","2","0"],["0","6","-10","0"],["0","3","2","-3"],["0","0","10","0"],["0","0","1","5"],["0","0","0","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-1","1","0"],["0","0","0","-4","0"],["0","0","0","-3","-3"],["0","0","0","0","-4"],["0","0","0","0","-1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":180692,"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-825960432a596355"]},{"id":180693,"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-825960432a596355"]},{"id":180694,"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-825960432a596355"]},{"id":180695,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-825960432a596355"]}],"region_exact":{},"incoming_edges":[{"id":"e-825960432a596355","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-0c8274243629c17a","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":"84a0e0c10c7f49c0ed4bbeb7c17ce3cf464b317aeee58d96ed24d5b869fad6ae","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-47fe395e9f7d5f86","role":"sequence","name":"T(n,k) is the number of partial bijections of height k (height(alpha) = |Im(alpha)|) of an n-element set.","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":[{"id":"l-b85f4215fc5334e8","kind":"fingerprint_related","node_a_id":"n-0c8274243629c17a","node_b_id":"n-b947e3321c316ff1","fingerprint":"f6-53219bd9184fb09a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-53219bd9184fb09a"},"other_id":"n-b947e3321c316ff1","other":{"id":"n-b947e3321c316ff1","name":"theta_plus_one_squared of Triangle read by rows: T(n, k) = 2^n * [x^k] hypergeom([-n, -n, 1], [1], x).","first_index":1,"fingerprint":"f6-53219bd9184fb09a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["4","64","72"],["8","288","1296","768"],["16","1024","10368","24576","9600"]],"card_rows":[["4","64","72"],["8","288","1296","768"],["16","1024","10368","24576","9600"],["32","3200","57600","307200","480000","138240"]],"row_sums_preview":["140","2360","45584","986272","23577280","616593280","17503289600","535865297408"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2ee8dc15b8ca5b2d","steps":["(\u03b8+1)\u00b2"],"edges":["e-c3a3342a9f442ff0"],"text":"A395972 \u2192 (\u03b8+1)\u00b2"},"headline":"A395972 \u2192 (\u03b8+1)\u00b2","chain":null}}],"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: 1,   4;\n  2: 1,  16,    18;\n  3: 1,  36,   162,     96;\n  4: 1,  64,   648,   1536,     600;\n  5: 1, 100,  1800,   9600,   15000,     4320;\n  6: 1, 144,  4050,  38400,  135000,   155520,     35280;\n  7: 1, 196,  7938, 117600,  735000,  1905120,   1728720,    322560;\n  8: 1, 256, 14112, 301056, 2940000, 13547520,  27659520,  20643840,   3265920;\n  9: 1, 324, 23328, 677376, 9525600, 68584320, 248935680, 418037760, 264539520, 36288000;\n","recurrence_sentence":"The row polynomials satisfy (1+(3*n^2-2*n)*x+n^4*x^2)*P(n,x) = (1+(3*n^2+6*n-4)*x+(n^4+10*n^3+2*n^2-10*n+2)*x^2+(2*n^5+5*n^4-3*n^2)*x^3)*P(n-1,x) - (x^2+(3*n^2+4*n-1)*x^3+(n^4+4*n^3+3*n^2)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((1 + (3*n^2 + 6*n - 4)*x + (n^4 + 10*n^3 + 2*n^2 - 10*n + 2)*x^2 + (2*n^5 + 5*n^4 - 3*n^2)*x^3)*P[n - 1, x] - (x^2 + (3*n^2 + 4*n - 1)*x^3 + (n^4 + 4*n^3 + 3*n^2)*x^4)*D[P[n - 1, x], x])/(1 + (3*n^2 - 2*n)*x + n^4*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((1 + (3*n^2 + 6*n - 4)*x + (n^4 + 10*n^3 + 2*n^2 - 10*n + 2)*x^2 + (2*n^5 + 5*n^4 - 3*n^2)*x^3)*v[i - 1] - (x^2 + (3*n^2 + 4*n - 1)*x^3 + (n^4 + 4*n^3 + 3*n^2)*x^4)*deriv(v[i - 1], x))/(1 + (3*n^2 - 2*n)*x + n^4*x^2)); 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-47fe395e9f7d5f86","steps":["(\u03b8+1)\u00b2"],"edges":["e-825960432a596355"],"text":"A144084 \u2192 (\u03b8+1)\u00b2"},"headline":"A144084 \u2192 (\u03b8+1)\u00b2","chain":null}
