{"id":"n-731ce0c1f40f7072","revision":0,"first_index":0,"fingerprint":"f6-f0acff1b7583ce22","name":"theta_plus_one of derivative of Triangle read by rows: T(n,k) is the number of Schroeder paths of semilength n containing exactly k peaks but no peaks at level one (n >= 1; 0 <= k <= n-1).","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"],["5","4"],["21","36","9"],["84","224","126","16"],["330","1200","1080","320","25"],["1287","5940","7425","3600","675","36"],["5005","28028","45045","30800","9625","1260","49"],["19448","128128","252252","224224","100100","22176","2156","64"],["75582","572832","1336608","1467648","859950","275184","45864","3456","81"],["293930","2519400","6802380","8910720","6497400","2751840","668850","87360","5265","100"],["1144066","10943240","33575850","51163200","44767800","23750496","7696920","1478400","155925","7700","121"],["4457400","47070144","161803620","281397600","287793000","184187520","75209904","19388160","3029400","264000","10890","144"],["17383860","200880160","764889840","1495784576","1752872550","1316940768","651819168","212838912","44895708","5834400","427856","14976","169"],["67863915","851809140","3559345335","7733886160","10225089875","8834477652","5153445297","2048574528","549972423","96996900","10669659","668304","20111","196"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","1","9","66","450","2955","18963","119812","748548","4637205","28537245","174683718","1064611782","6464582943","39132819495"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["1"]],"recurrence":{"denominator":{"coeffs":[["-2","0"],["-3","2"],["3","-3"],["2","0"],["0","1"]]},"terms":[{"coeff":{"coeffs":[["0","-4","2"],["2","-10","-1"],["8","0","-4"],["8","10","0"],["0","4","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-1","0","1"],["0","-4","-4","0"],["0","-4","-6","-2"],["0","0","-2","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":177435,"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-83b6259d16c6e61c"]},{"id":177436,"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-83b6259d16c6e61c"]},{"id":177437,"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-83b6259d16c6e61c"]}],"region_exact":{},"incoming_edges":[{"id":"e-83b6259d16c6e61c","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-731ce0c1f40f7072","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":"fb00dcf56da4165d5c3bcf257f0141cfc3b9491b6afeb4c1f8b19a3301beed61","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-3e16b3cbabac7254","role":"sequence","name":"derivative of Triangle read by rows: T(n,k) is the number of Schroeder paths of semilength n containing exactly k peaks but no peaks at level one (n >= 1; 0 <= k <= n-1).","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-35de83d0525ee14d","kind":"fingerprint_related","node_a_id":"n-731ce0c1f40f7072","node_b_id":"n-398a4845fac4d9f1","fingerprint":"f6-f0acff1b7583ce22","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-f0acff1b7583ce22"},"other_id":"n-398a4845fac4d9f1","other":{"id":"n-398a4845fac4d9f1","name":"theta_plus_one of basis_map_A007318 of Triangle of Dyck paths counted by number of long interior inclines.","first_index":0,"fingerprint":"f6-f0acff1b7583ce22","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["2"],["5","4"],["14","24","6"]],"card_rows":[["2"],["5","4"],["14","24","6"],["42","112","63","8"]],"row_sums_preview":["2","9","44","225","1182","6321","34232","187137"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c963abfc0266d66c","steps":["x \u21a6 x+1","\u03b8+1"],"edges":["e-454321f2b34b0823","e-47d1189af3bc51cd"],"text":"A108838 \u2192 x \u21a6 x+1 \u2192 \u03b8+1"},"headline":"A108838 \u2192 x \u21a6 x+1 \u2192 \u03b8+1","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:     0;\n  1:     1;\n  2:     5,      4;\n  3:    21,     36,       9;\n  4:    84,    224,     126,      16;\n  5:   330,   1200,    1080,     320,     25;\n  6:  1287,   5940,    7425,    3600,    675,     36;\n  7:  5005,  28028,   45045,   30800,   9625,   1260,    49;\n  8: 19448, 128128,  252252,  224224, 100100,  22176,  2156,   64;\n  9: 75582, 572832, 1336608, 1467648, 859950, 275184, 45864, 3456, 81;\n","recurrence_sentence":"The row polynomials satisfy (2*n^3+3*n^2-3*n-2+(n^4-3*n^2+2*n)*x)*P(n,x) = (8*n^3+8*n^2+2*n+(4*n^4+10*n^3-10*n-4)*x+(3*n^4-4*n^2-n+2)*x^2)*P(n-1,x) - ((4*n^2+4*n+1)*x+(2*n^3+6*n^2+4*n)*x^2+(2*n^3+2*n^2-1)*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((8*n^3 + 8*n^2 + 2*n + (4*n^4 + 10*n^3 - 10*n - 4)*x + (3*n^4 - 4*n^2 - n + 2)*x^2)*P[n - 1, x] - ((4*n^2 + 4*n + 1)*x + (2*n^3 + 6*n^2 + 4*n)*x^2 + (2*n^3 + 2*n^2 - 1)*x^3)*D[P[n - 1, x], x])/(2*n^3 + 3*n^2 - 3*n - 2 + (n^4 - 3*n^2 + 2*n)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((8*n^3 + 8*n^2 + 2*n + (4*n^4 + 10*n^3 - 10*n - 4)*x + (3*n^4 - 4*n^2 - n + 2)*x^2)*v[i - 1] - ((4*n^2 + 4*n + 1)*x + (2*n^3 + 6*n^2 + 4*n)*x^2 + (2*n^3 + 2*n^2 - 1)*x^3)*deriv(v[i - 1], x))/(2*n^3 + 3*n^2 - 3*n - 2 + (n^4 - 3*n^2 + 2*n)*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-57b4b14604f4c0d7","steps":["D","\u03b8+1"],"edges":["e-a4f714b61c97080d","e-83b6259d16c6e61c"],"text":"A126216 \u2192 D \u2192 \u03b8+1"},"headline":"A126216 \u2192 D \u2192 \u03b8+1","chain":null}
