{"id":"n-278fa84b7ef5cf2b","revision":0,"first_index":0,"fingerprint":"f6-ab1b08e67af73826","name":"theta_plus_one_squared of gamma_polynomial of Triangle T(n, k) of numbers of square lattice walks that start and end at origin after 2*n steps and contain exactly k steps to the east, possibly touching origin at intermediate stages.","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","48"],["20","480"],["70","3360","3780"],["252","20160","68040"],["924","110880","748440","295680"],["3432","576576","6486480","7687680"],["12870","2882880","48648600","115315200","22522500"],["48620","14002560","330810480","1306905600","765765000"],["184756","66512160","2095133040","12415603200","14549535000","1676106432"],["705432","310390080","12570798240","104291066880","203693490000","70396470144"],["2704156","1427794368","72282089880","799564846080","2342475135000","1619118813312","122433367056"],["10400600","6489974400","401567166000","5711177472000","23424751350000","26985313555200","6121668352800"],["40116600","29204884800","2168462696400","38550447936000","210822762150000","364301732995200","165285045525600","8811530956800"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","2","54","500","7210","88452","1155924","14754168","189382050","2417532260","30803074588","391262920776","4957304749852","62650978271000","789969227261400"],"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":[["0","0","0"],["0","0","0"],["1","2","0"],["0","0","0"],["0","-4","1"],["0","0","2"],["0","0","1"]]},"terms":[{"coeff":{"coeffs":[["0","32","-16","32"],["-2","-100","-56","0"],["4","72","224","-120"],["0","8","-42","-72"],["0","-16","-112","72"],["0","0","6","72"],["0","0","4","16"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-4","24","-48","64"],["0","8","-16","-80","64"],["0","0","-48","236","-176"],["0","0","-32","208","-320"],["0","0","0","44","-176"],["0","0","0","8","-32"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":184065,"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-1ecb280c35194aee"]},{"id":184066,"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-1ecb280c35194aee"]},{"id":184067,"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-1ecb280c35194aee"]}],"region_exact":{},"incoming_edges":[{"id":"e-1ecb280c35194aee","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-278fa84b7ef5cf2b","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":"9bba737d66ed2bd67a578598a41d1b81f1c5e15e9412bc632636794ff32d5f08","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-ce7f61ca1eb91390","role":"sequence","name":"gamma_polynomial of Triangle T(n, k) of numbers of square lattice walks that start and end at origin after 2*n steps and contain exactly k steps to the east, possibly touching origin at intermediate stages.","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-10630808b6a0e989","kind":"fingerprint_related","node_a_id":"n-15c46876416f2009","node_b_id":"n-278fa84b7ef5cf2b","fingerprint":"f6-ab1b08e67af73826","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-ab1b08e67af73826"},"other_id":"n-15c46876416f2009","other":{"id":"n-15c46876416f2009","name":"theta_plus_one_squared of T(n,k) = binomial(n,2*k)*binomial(2*k,k) for 0 <= k <= n, triangle read by rows.","first_index":1,"fingerprint":"f6-ab1b08e67af73826","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["1","8"]],"card_rows":[["1"],["1"],["1","8"],["1","24"]],"row_sums_preview":["1","1","9","25","103","351","1251","4299"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4b168caf0bc63f8b","steps":["(\u03b8+1)\u00b2"],"edges":["e-0345e656c1a770ab"],"text":"A089627 \u2192 (\u03b8+1)\u00b2"},"headline":"A089627 \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-685cbd9b0679727e","kind":"fingerprint_related","node_a_id":"n-906c3033a0728ce4","node_b_id":"n-278fa84b7ef5cf2b","fingerprint":"f6-ab1b08e67af73826","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-ab1b08e67af73826"},"other_id":"n-906c3033a0728ce4","other":{"id":"n-906c3033a0728ce4","name":"theta_plus_one_squared of Triangle read by rows: T(n,k) is k-th entry of the gamma-vector of the n-dimensional cyclohedron, 0 <= k <= floor(n/2).","first_index":0,"fingerprint":"f6-ab1b08e67af73826","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["1","8"]],"card_rows":[["1"],["1"],["1","8"],["1","24"]],"row_sums_preview":["1","1","9","25","103","351","1251","4299"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8bfdd58279be42d7","steps":["(\u03b8+1)\u00b2"],"edges":["e-7d4659a9e7a9f883"],"text":"A391403 \u2192 (\u03b8+1)\u00b2"},"headline":"A391403 \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:     2;\n  2:     6,       48;\n  3:    20,      480;\n  4:    70,     3360,      3780;\n  5:   252,    20160,     68040;\n  6:   924,   110880,    748440,     295680;\n  7:  3432,   576576,   6486480,    7687680;\n  8: 12870,  2882880,  48648600,  115315200,  22522500;\n  9: 48620, 14002560, 330810480, 1306905600, 765765000;\n","recurrence_sentence":"The row polynomials satisfy (n^2-(4*n^4-2*n^2)*x+(n^6+2*n^5+n^4)*x^2)*P(n,x) = (4*n^2-2*n-(16*n^4-8*n^3-72*n^2+100*n-32)*x+(4*n^6+6*n^5-112*n^4-42*n^3+224*n^2-56*n-16)*x^2+(16*n^6+72*n^5+72*n^4-72*n^3-120*n^2+32)*x^3)*P(n-1,x) + ((8*n-4)*x-(32*n^3+48*n^2+16*n-24)*x^2+(8*n^5+44*n^4+208*n^3+236*n^2-80*n-48)*x^3-(32*n^5+176*n^4+320*n^3+176*n^2-64*n-64)*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[((4*n^2 - 2*n - (16*n^4 - 8*n^3 - 72*n^2 + 100*n - 32)*x + (4*n^6 + 6*n^5 - 112*n^4 - 42*n^3 + 224*n^2 - 56*n - 16)*x^2 + (16*n^6 + 72*n^5 + 72*n^4 - 72*n^3 - 120*n^2 + 32)*x^3)*P[n - 1, x] + ((8*n - 4)*x - (32*n^3 + 48*n^2 + 16*n - 24)*x^2 + (8*n^5 + 44*n^4 + 208*n^3 + 236*n^2 - 80*n - 48)*x^3 - (32*n^5 + 176*n^4 + 320*n^3 + 176*n^2 - 64*n - 64)*x^4)*D[P[n - 1, x], x])/(n^2 - (4*n^4 - 2*n^2)*x + (n^6 + 2*n^5 + 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] = ((4*n^2 - 2*n - (16*n^4 - 8*n^3 - 72*n^2 + 100*n - 32)*x + (4*n^6 + 6*n^5 - 112*n^4 - 42*n^3 + 224*n^2 - 56*n - 16)*x^2 + (16*n^6 + 72*n^5 + 72*n^4 - 72*n^3 - 120*n^2 + 32)*x^3)*v[i - 1] + ((8*n - 4)*x - (32*n^3 + 48*n^2 + 16*n - 24)*x^2 + (8*n^5 + 44*n^4 + 208*n^3 + 236*n^2 - 80*n - 48)*x^3 - (32*n^5 + 176*n^4 + 320*n^3 + 176*n^2 - 64*n - 64)*x^4)*deriv(v[i - 1], x))/(n^2 - (4*n^4 - 2*n^2)*x + (n^6 + 2*n^5 + 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-fdc03fad975441d0","steps":["\u03b3","(\u03b8+1)\u00b2"],"edges":["e-510754e51c3ada00","e-1ecb280c35194aee"],"text":"A069466 \u2192 \u03b3 \u2192 (\u03b8+1)\u00b2"},"headline":"A069466 \u2192 \u03b3 \u2192 (\u03b8+1)\u00b2","chain":null}