{"id":"n-4cab0236cf9d2d4f","revision":0,"first_index":1,"fingerprint":"f6-2c1fea4a558d4e6c","name":"derivative of Triangle read by rows: T(n,k) = n!*(n+k-1)!/((n-k)!*(n-1)!*(k!)^2) for 0 <= k <= n, with T(0,0) = 1.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[[],["1"],["4","6"],["9","36","30"],["16","120","240","140"],["25","300","1050","1400","630"],["36","630","3360","7560","7560","2772"],["49","1176","8820","29400","48510","38808","12012"],["64","2016","20160","92400","221760","288288","192192","51480"],["81","3240","41580","249480","810810","1513512","1621620","926640","218790"],["100","4950","79200","600600","2522520","6306300","9609600","8751600","4375800","923780"],["121","7260","141570","1321320","6936930","22198176","44924880","57760560","45727110","20323160","3879876"],["144","10296","240240","2702700","17297280","68612544","176432256","299304720","332560800","232792560","93117024","16224936"],["169","14196","390390","5205200","39819780","191134944","605260656","1296987120","1891439550","1849407560","1160082924","421848336","67603900"],["196","19110","611520","9529520","85765680","488864376","1862340480","4888643760","8962513560","11452100660","9994560576","5678727600","1892909200","280816200"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","1","10","75","516","3405","21918","138775","868360","5385753","33174450","203220963","1239295500","7529194725","45597402438"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["0"],["1"],["4","6"],["9","36","30"]],"recurrence":{"denominator":{"coeffs":[["126"],["-253"],["167"],["-44"],["4"]]},"terms":[{"coeff":{"coeffs":[["-300"],["455"],["-239"],["52"],["-4"]]},"deriv_order":0,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["1056","2160"],["-1640","-3324"],["888","1784"],["-200","-400"],["16","32"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["-1350","-3672","-3780"],["2172","5856","5952"],["-1226","-3280","-3296"],["288","768","768"],["-24","-64","-64"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["720","1512"],["-1240","-2532"],["744","1496"],["-184","-368"],["16","32"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":177292,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-5b009bca5501b390"]},{"id":177293,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-5b009bca5501b390"]},{"id":177294,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-5b009bca5501b390"]}],"region_exact":{},"incoming_edges":[{"id":"e-5b009bca5501b390","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"0\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-4cab0236cf9d2d4f","premise_count":1,"theorem":"Laguerre's theorem on differential operators: if phi is a real polynomial with only real roots, then phi(D) maps every real-rooted polynomial to a real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"cc71483b8732aedb5ddc3286519e7886cc885381658703693cd817dd93a3e376","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-55697ddae4d22d60","role":"sequence","name":"Triangle read by rows: T(n,k) = n!*(n+k-1)!/((n-k)!*(n-1)!*(k!)^2) for 0 <= k <= n, with T(0,0) = 1.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-1f9afd2ea3aa20b5","kind":"fingerprint_related","node_a_id":"n-c2b0d68b9e568dde","node_b_id":"n-4cab0236cf9d2d4f","fingerprint":"f6-2c1fea4a558d4e6c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2c1fea4a558d4e6c"},"other_id":"n-c2b0d68b9e568dde","other":{"id":"n-c2b0d68b9e568dde","name":"basis_map_A007318 of Triangle read by rows. T(n, k) = binomial(n, k)*binomial(n - 1, n - k - 1).","first_index":0,"fingerprint":"f6-2c1fea4a558d4e6c","polytool_family":null,"degree_pattern":"irregular","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["3","2"]],"card_rows":[["1"],["1"],["3","2"],["10","12","3"]],"row_sums_preview":["1","1","5","25","129","681","3653","19825"],"regions":["(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-b1b19325e0094118","steps":["x \u21a6 x+1"],"edges":["e-0de802f77814f187"],"text":"A367270 \u2192 x \u21a6 x+1"},"headline":"A367270 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-82795db91991d90f","kind":"fingerprint_related","node_a_id":"n-a60016fa379da581","node_b_id":"n-4cab0236cf9d2d4f","fingerprint":"f6-2c1fea4a558d4e6c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2c1fea4a558d4e6c"},"other_id":"n-a60016fa379da581","other":{"id":"n-a60016fa379da581","name":"shift_x_plus_one of theta_plus_one of Triangle of Narayana numbers T(n,k) = C(n-1,k-1)*C(n,k-1)/k with 1 <= k <= n, read by rows. Also called the Catalan triangle.","first_index":0,"fingerprint":"f6-2c1fea4a558d4e6c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["3","2"],["10","12","3"]],"card_rows":[["1"],["3","2"],["10","12","3"],["35","60","30","4"]],"row_sums_preview":["1","5","25","129","681","3653","19825","108545"],"regions":["(-inf, -1]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c2eaf5212a4dec13","steps":["\u03b8+1","x \u21a6 x+1"],"edges":["e-127d139330d3e4d4","e-09282c699b4099b7"],"text":"A001263 \u2192 \u03b8+1 \u2192 x \u21a6 x+1"},"headline":"A001263 \u2192 \u03b8+1 \u2192 x \u21a6 x+1","chain":null}},{"id":"l-9ebf088d8c03ed23","kind":"fingerprint_related","node_a_id":"n-43c5f46a251ccde6","node_b_id":"n-4cab0236cf9d2d4f","fingerprint":"f6-2c1fea4a558d4e6c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2c1fea4a558d4e6c"},"other_id":"n-43c5f46a251ccde6","other":{"id":"n-43c5f46a251ccde6","name":"reversal of Triangle read by rows. T(n, k) = (1/2) * binomial(2*(n - k + 1), n - k + 1) * binomial(2*n - k, k - 1) for n > 0, T(0, 0) = 1.","first_index":0,"fingerprint":"f6-2c1fea4a558d4e6c","polytool_family":null,"degree_pattern":"irregular","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["2","3"]],"card_rows":[["1"],["1"],["2","3"],["3","12","10"]],"row_sums_preview":["1","1","5","25","129","681","3653","19825"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-0a5e54eacdb24124","steps":["reverse"],"edges":["e-a6f5a3bfa125593b"],"text":"A360282 \u2192 reverse"},"headline":"A360282 \u2192 reverse","chain":null}},{"id":"l-b93ef10cd246ff88","kind":"fingerprint_related","node_a_id":"n-0a5e54eacdb24124","node_b_id":"n-4cab0236cf9d2d4f","fingerprint":"f6-2c1fea4a558d4e6c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2c1fea4a558d4e6c"},"other_id":"n-0a5e54eacdb24124","other":{"id":"n-0a5e54eacdb24124","name":"Triangle read by rows. T(n, k) = (1/2) * binomial(2*(n - k + 1), n - k + 1) * binomial(2*n - k, k - 1) for n > 0, T(0, 0) = 1.","first_index":0,"fingerprint":"f6-2c1fea4a558d4e6c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["0","1"],["0","3","2"]],"card_rows":[["1"],["0","1"],["0","3","2"],["0","10","12","3"]],"row_sums_preview":["1","1","5","25","129","681","3653","19825"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":["A360282"],"derivation":{"root":"n-0a5e54eacdb24124","steps":[],"edges":[],"text":"A360282"},"headline":"A360282: Triangle read by rows. T(n, k) = (1/2) * binomial(2*(n - k + 1), n - k + 1) * binomial(2*n - k, k - 1) for n > 0, T(0, 0) = 1.","chain":null}},{"id":"l-e5b65275698d943d","kind":"fingerprint_related","node_a_id":"n-4cab0236cf9d2d4f","node_b_id":"n-133f3a16dadf73ac","fingerprint":"f6-2c1fea4a558d4e6c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2c1fea4a558d4e6c"},"other_id":"n-133f3a16dadf73ac","other":{"id":"n-133f3a16dadf73ac","name":"reversal of Triangle read by rows: coefficients of the partial fraction decomposition of [d^n/dx^n] (x/(1-x))^n/n!.","first_index":1,"fingerprint":"f6-2c1fea4a558d4e6c","polytool_family":null,"degree_pattern":"irregular","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["3","2"]],"card_rows":[["1"],["1"],["3","2"],["10","12","3"]],"row_sums_preview":["1","1","5","25","129","681","3653","19825"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-b0d6ec97a8d172b8","steps":["reverse"],"edges":["e-70add7cce9667674"],"text":"A253283 \u2192 reverse"},"headline":"A253283 \u2192 reverse","chain":null}},{"id":"l-f359266cdcf70e49","kind":"fingerprint_related","node_a_id":"n-b0d6ec97a8d172b8","node_b_id":"n-4cab0236cf9d2d4f","fingerprint":"f6-2c1fea4a558d4e6c","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-2c1fea4a558d4e6c"},"other_id":"n-b0d6ec97a8d172b8","other":{"id":"n-b0d6ec97a8d172b8","name":"Triangle read by rows: coefficients of the partial fraction decomposition of [d^n/dx^n] (x/(1-x))^n/n!.","first_index":1,"fingerprint":"f6-2c1fea4a558d4e6c","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["0","1"],["0","2","3"]],"card_rows":[["1"],["0","1"],["0","2","3"],["0","3","12","10"]],"row_sums_preview":["1","1","5","25","129","681","3653","19825"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":["A253283"],"derivation":{"root":"n-b0d6ec97a8d172b8","steps":[],"edges":[],"text":"A253283"},"headline":"A253283: Triangle read by rows: coefficients of the partial fraction decomposition of [d^n/dx^n] (x/(1-x))^n/n!.","chain":null}}],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:  0;\n   2:  1;\n   3:  4,    6;\n   4:  9,   36,    30;\n   5: 16,  120,   240,    140;\n   6: 25,  300,  1050,   1400,    630;\n   7: 36,  630,  3360,   7560,   7560,    2772;\n   8: 49, 1176,  8820,  29400,  48510,   38808,   12012;\n   9: 64, 2016, 20160,  92400, 221760,  288288,  192192,  51480;\n  10: 81, 3240, 41580, 249480, 810810, 1513512, 1621620, 926640, 218790;\n","recurrence_sentence":"The row polynomials satisfy (4*n^4-44*n^3+167*n^2-253*n+126)*P(n,x) = -(4*n^4-52*n^3+239*n^2-455*n+300)*P(n-4,x) + (16*n^4-200*n^3+888*n^2-1640*n+1056+(32*n^4-400*n^3+1784*n^2-3324*n+2160)*x)*P(n-3,x) - (24*n^4-288*n^3+1226*n^2-2172*n+1350+(64*n^4-768*n^3+3280*n^2-5856*n+3672)*x+(64*n^4-768*n^3+3296*n^2-5952*n+3780)*x^2)*P(n-2,x) + (16*n^4-184*n^3+744*n^2-1240*n+720+(32*n^4-368*n^3+1496*n^2-2532*n+1512)*x)*P(n-1,x) for n >= 5, with P(1,x) = 0, P(2,x) = 1, P(3,x) = 4 + 6*x, P(4,x) = 9 + 36*x + 30*x^2.","mathematica":"P[1, x_] := 0;\nP[2, x_] := 1;\nP[3, x_] := 4 + 6*x;\nP[4, x_] := 9 + 36*x + 30*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(4*n^4 - 52*n^3 + 239*n^2 - 455*n + 300)*P[n - 4, x] + (16*n^4 - 200*n^3 + 888*n^2 - 1640*n + 1056 + (32*n^4 - 400*n^3 + 1784*n^2 - 3324*n + 2160)*x)*P[n - 3, x] - (24*n^4 - 288*n^3 + 1226*n^2 - 2172*n + 1350 + (64*n^4 - 768*n^3 + 3280*n^2 - 5856*n + 3672)*x + (64*n^4 - 768*n^3 + 3296*n^2 - 5952*n + 3780)*x^2)*P[n - 2, x] + (16*n^4 - 184*n^3 + 744*n^2 - 1240*n + 720 + (32*n^4 - 368*n^3 + 1496*n^2 - 2532*n + 1512)*x)*P[n - 1, x])/(4*n^4 - 44*n^3 + 167*n^2 - 253*n + 126)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 1; v[3] = 4 + 6*x; v[4] = 9 + 36*x + 30*x^2; for(n = 5, N, my(i = n); v[i] = (-(4*n^4 - 52*n^3 + 239*n^2 - 455*n + 300)*v[i - 4] + (16*n^4 - 200*n^3 + 888*n^2 - 1640*n + 1056 + (32*n^4 - 400*n^3 + 1784*n^2 - 3324*n + 2160)*x)*v[i - 3] - (24*n^4 - 288*n^3 + 1226*n^2 - 2172*n + 1350 + (64*n^4 - 768*n^3 + 3280*n^2 - 5856*n + 3672)*x + (64*n^4 - 768*n^3 + 3296*n^2 - 5952*n + 3780)*x^2)*v[i - 2] + (16*n^4 - 184*n^3 + 744*n^2 - 1240*n + 720 + (32*n^4 - 368*n^3 + 1496*n^2 - 2532*n + 1512)*x)*v[i - 1])/(4*n^4 - 44*n^3 + 167*n^2 - 253*n + 126)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-55697ddae4d22d60","steps":["D"],"edges":["e-5b009bca5501b390"],"text":"A123160 \u2192 D"},"headline":"A123160 \u2192 D","chain":null}
